82770969.pdf

7/28/2019 82770969.pdf

1/37

The Optimal Ination Rate inNew Keynesian Models: Should

Central Banks Raise TheirInation Targets in Light o the

Zero Lower Bound?OLIVIER COIBION

College o William and Mary and NBER

YURIY GORODNICHENKOUniversity o Caliornia Berkeley, NBER, and IZA

and

JOHANNES WIELANDUniversity o Caliornia Berkeley

First version received March 2011; fnal version accepted December2011 (Eds .)

We study the eects o positive steady-state ination in New Keynesian models subject to the zerobound on interest rates. We derive the utility-based welare loss unction taking into account the eects

o positive steady-state ination and solve or the optimal level o ination in the model. For plausible

calibrations with costly but inrequent episodes at the zero lower bound, the optimal ination rate is low,

typically

7/28/2019 82770969.pdf

2/37

REVIEW OF ECONOMIC STUDIES

1. INTRODUCTION

One o the defning eatures o the current economic crisis has been the zero bound on nominal

interest rates. With standard monetary policy running out o ammunition in the midst o one othe sharpest downturns in post-World War II economic history, some have suggested that central

banks should consider allowing or higher target ination rates than would have been consid-

ered reasonable just a ew years ago. We contribute to this question by explicitly incorporatingpositive steady-state (or trend) ination into New Keynesian models as well as the zero lowerbound (ZLB) on nominal interest rates. We derive the eects o non-zero steady-state ination

on the loss unction, thereby laying the groundwork or welare analysis. While hitting the ZLB

is very costly in the model, our baseline fnding is that the optimal rate o ination is low, typ-

ically

7/28/2019 82770969.pdf

3/37

COIBION ET AL. OPTIMAL INFLATION

episodes, as suggested by Blanchard in the opening quote. Our approach or modelling the zerobound ollows Bodenstein, Erceg and Guerrieri (2009) by solving or the duration o the zero

bound endogenously, unlike in Christiano, Eichenbaum and Rebelo (2011) or Eggertsson and

Woodord (2004). This is important because the welare costs o ination are a unction o the

variance o ination and output, which themselves depend on the requency at which the zero

bound is reached as well as the duration o zero bound episodes.Ater calibrating the model to broadly match the moments o macro-economic series and

the historical incidence o hitting the ZLB in the U.S., we then solve or the rate o ination

that maximizes welare. While the ZLB ensures that the optimal ination rate is positive, or

plausible calibrations o the structural parameters o the model and the properties o the shocks

driving the economy, the optimal ination rate is quite low: typically

7/28/2019 82770969.pdf

4/37


ination rate rom 15% to 19% per year. Second, one might be concerned that our fndingshinge on modelling price stickiness as in Calvo (1983). Since this approach implies that some

frms do not change prices or extended periods o time, it could overstate the cost o price disper-

sion and thereore understate the optimal ination rate. To address this possibility, we reproduce

our analysis using Taylor (1977) staggered price setting o fxed durations. The latter reduces

price dispersion relative to the Calvo assumption but raises the optimal ination rate to only22% when prices are changed every three quarters. Another limitation o the Calvo assumption

is that the rate at which prices are changed is commonly treated as a structural parameter, yet the

requency o price setting may depend on the ination rate even or low ination rates like those

experienced in the U.S. As a result, we consider two modifcations that allow or price exibility

to vary with the trend rate o ination. In the frst specifcation, we let the degree o price rigidityvary systematically with the trend level o ination. In the second specifcation, we employ the

Dotsey, King and Wolman (1999) model o state-dependent pricing, which allows the degree

o price stickiness to vary endogenously both in the short-run and in the long-run, and thus we

address one o the major criticisms o the previous literature on the optimal ination rate. Both

modifcations yield optimal ination rates o

7/28/2019 82770969.pdf

5/37


ination in the context o New Keynesian models. However, their calibration implies that thechance o hitting the ZLB is practically zero and thereore does not quantitatively aect the

optimal rate o ination, whereas we ocus on a setting where costly ZLB events occur at their

historic requency. Furthermore, none o these papers consider the endogenous nature o price

rigidity with respect to trend ination.

An advantage o working with a micro-ounded model and its implied welare unction isthe ability to engage in normative analysis. In our baseline model, the endogenous response o

monetary policymakers to macro-economic conditions is captured by a Taylor rule. Thus, we

are also able to study the welare eects o altering the systematic response o policymakers to

endogenous uctuations (i.e. the coefcients o the Taylor rule) and determine the new optimal

steady-state rate o ination. The most striking fnding rom this analysis is that even modestprice-level targeting (PLT) would raise welare by non-trivial amounts or any steady-state ina-

tion rate and come close to the Ramsey-optimal policy, consistent with the fnding oEggertsson

and Woodord (2003) and Wolman (2005). In short, the optimal policy rule or the model can be

closely characterized by the name o price stability as typically stated in the legal mandates o

most central banks.Given our results, we conclude that raising the target rate o ination is likely too blunt an

instrument to reduce the incidence and severity o zero bound episodes. In all the New Keynesianmodels we consider, even the small costs associated with higher trend ination rates, which

must be borne every period, more than oset the welare benefts o ewer and less severe ZLB

events. Instead, changes in the policy rule, such as PLT, may be more eective both in avoidingand minimizing the costs associated with these crises. In the absence o such changes to the

interest rate rule, our results suggest that addressing the large welare losses associated with the

ZLB is likely to best be pursued through policies targeted specifcally to these episodes, such as

countercyclical fscal policy or the use o non-standard monetary policy tools.

Section 2 presents the baseline New Keynesian model and derivations when allowing orpositive steady-state ination, including the associated loss unction. Section 3 includes our

calibration o the model as well as the results or the optimal rate o ination, while Section 4

investigates the robustness o our results to parameter values. Section 5 then considers extensionso the baseline model which could potentially lead to higher estimates o the optimal ination

target. Section 6 considers additional normative implications o the model, including optimalstabilization policy and PLT. Section 7 concludes.

2. A NEW KEYNESIAN MODEL WITH POSITIVE STEADY-STATE INFLATION

We consider a standard New Keynesian model with a representative consumer, a continuum o

monopolistic producers o intermediate goods, a fscal authority, and a central bank.

2.1. Model

The representative consumer maximizes the present discounted value o the utility stream romconsumption and leisure

maxEt

j =0

j

log(Ct+j hG At+j Ct+j 1) + 11

0

Nt+j (i )1+1/ di

, (1)where C is consumption o the fnal good, N(i ) is labour supplied to individual industry i , G A

is the gross growth rate o technology, is the Frisch labour supply elasticity, h the internal habit

1375
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

6/37


parameter, and is the discount actor.2 The budget constraint each period t is given by

t: Ct + St/Pt

10

Nt(i )Wt(i)/Pt

di + St1qt1Rt1/Pt + Tt, (2)

where S is the stock o one-period bonds held by the consumer, R is the gross nominal interest

rate, P is the price o the fnal good, W(i) is the nominal wage earned rom labour in industry i ,

T is real transers and profts rom ownership o frms, q is a risk premium shock, and is the

shadow value o wealth.3 The frst-order conditions rom this utility-maximization problem arethen:

(Ct hG AtCt1)1 h EtG At+1(Ct+1 hG At+1Ct)

1 = t, (3)

Nt(i)1/ = tWt(i)/Pt, (4)

t/Pt =Et[t+1qtRt/Pt+1]. (5)

Production o the fnal good is done by a perectly competitive sector that combines a

continuum o intermediate goods into a fnal good per the ollowing aggregator:

Yt = 10

Yt(i)(1)/ di

/(1)

, (6)

where Y is the fnal good and Y(i) is intermediate good i , while denotes the elasticity o substi-

tution across intermediate goods, yielding the ollowing demand curve or goods o intermediate

sector i :Yt(i ) = Yt(Pt(i )/Pt)

(7)

and the ollowing expression or the aggregate price level:

Pt =

1

0Pt(i)

(1)di

1/(1)

. (8)

The production o each intermediate good is done by a monopolist acing a production unc-tion linear in labour

Yt(i) = AtNt(i ), (9)

where A denotes the level o technology, common across frms. Each intermediate good producerhas sticky prices, modelled as in Calvo (1983), where 1 is the probability that each frm will

be able to re-optimize its price each period. We allow or indexation o prices to steady-state

ination by frms who do not re-optimize their prices each period, with representing the degree

o indexation (0 or no indexation to 1 or ull indexation). Denoting the optimal reset price o

frm i by B(i), re-optimizing frms solve the ollowing proft-maximization problem:

maxEt

j =0

j Q t,t+j Yt+j (i )Bt(i)j Wt+j (i )Nt+j (i ), (10)2. We use internal habits rather than external habits because they more closely match the (lack o) persistence in

consumption growth in the data. The gross growth rate o technology enters the habit term to simpliy derivations.

3. As discussed in Smets and Wouters (2007), a positive shock to q , which is the wedge between the interest rate

controlled by the central bank and the return on assets held by the households, increases the required return on assets and

reduces current consumption. The shockq has similar eects as net-worth shocks in models with fnancial accelerators.

Such fnancial shocks have arguably played a major role in causing the ZLB to bind in practice. Amano and Shukayev

(2010) also document that shocks like q are essential or generating a binding ZLB in the New Keynesian model.

1376
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

7/37


where Q is the stochastic discount actor and is the gross steady-state level o ination. Theoptimal relative reset price is then given by

Bt(i)

Pt=

1

Et

j =0 j Qt,t+j Yt+j (Pt+j /Pt)

+1 j MCt+j (i)/Pt+j

Et

j =0

j Qt,t+j Yt+j (Pt+j /Pt) j (1), (11)

where frm-specifc marginal costs can be related to aggregate variables using

MCt+j (i )

Pt+j=

1t+j

At+j

Yt+j

At+j

1/Bt(i)

Pt

/ Pt+j j Pt

/. (12)

Given these price-setting assumptions, the dynamics o the price level are governed by

P1t = (1 )B1t + P

1t1

(1 ). (13)

We allow or government consumption o fnal goods (G), so the goods market clearing

condition or the economy is

Yt = Ct + G t. (14)We defne the aggregate labour input as

Nt =

10

Nt(i )(1)/di

/(1). (15)

2.2. Steady state and log-linearization

Following Coibion and Gorodnichenko (2011), we log-linearize the model around the

steady-state in which ination need not be zero. Since positive trend ination may imply that

the steady-state and the exible-price level o output dier, we adopt the ollowing notational

convention. Variables with a bar denote steady-state values, or example, Y t is the steady-statelevel o output. Lower-case letters denote the log o a variable, or example, yt = log Yt is the

log o current output. We assume that technology is a random walk and hence we normalize all

non-stationary real variables by the level o technology. We let hats on lower-case letters de-

note log-deviations rom steady-state, or example, yt = yt yt is the approximate percentage

deviation o output rom steady-state. Since we defne the steady-state as embodying the currentlevel o technology, deviations rom the steady-state are stationary. Finally, we denote deviations

rom the exible-price level steady-state with a tilde, or example, yt = yt yFt is the approxi-

mate percentage deviation o output rom its exible-price steady-state, where the superscript F

denotes a exible-price level quantity. Defne the net steady-state level o ination as =log(). The log-linearized consumption Euler equation is

t = Ett+1 + rt t+1 + qt, (16)where the marginal utility o consumption is given by

t =h

(1 h) (1 h)ct1

1 h2

(1 h) (1 h)ct +

h

(1 h) (1 h)Etct+1

and the goods market clearing condition becomes

yt = cy ct + gy gt , (17)

1377
http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

8/37


where cy and gy are the steady-state ratios o consumption and government to output, respec-tively. Also, integrating over frm-specifc production unctions and log-linearizing yields

yt = nt . (18)

Allowing or positive steady-state ination (i.e. > 0) primarily aects the steady-state and

price-setting components o the model. For example, the steady-state level o the output gap(which is defned as the deviation o steady-state output rom its exible-price level counterpart

Xt = Yt /YF

t ) is given by

X(+1)/ =1 (1)(+1)/

1 (1)(1)

1

1 (1)(1)

(+)/((1)). (19)

Note that the steady-state level o the gap is equal to one when steady-state ination is zero(i.e. = 1) or when the degree o price indexation is exactly equal to one. As emphasized by

Ascari and Ropele (2007), there is a non-linear relationship between the steady-state levels o

ination and output. For very low but positive trend ination, X is increasing in trend ination

but the sign is quickly reversed so that X is alling with trend ination or most positive levelso trend ination.

Secondly, positive steady-state ination aects the relationship between aggregate ination

and the re-optimizing price. Specifcally, the relationship between the two in the steady state is

now given by

(B/P) =

1

1 (1)(1)

1/(1), (20)

and the log-linearized equation is described by

t = 1 (1)(1)(1)(1)

bt bt = Mt, (21)so that ination is less sensitive to changes in the re-optimizing price as steady-state inationrises because goods with high relative prices receive a smaller share o expenditures.

Similarly, positive steady-state ination has important eects on the log-linearized optimal

reset price equation, which is given by1 +

bt = (1 2)

j =0

j

2

1

Et yt+j Ett+j

+Et

j =1

(j

2 j

1 )(gy t+j + rt+j 1)+

j =1 j2 1 + ( + 1) j1 Ett+j +m t, (22)

where m t is a cost-push shock, 1 = (1)(1), and 2 = 1(1)(1+/) so that without

steady-state ination or ull indexation, we have 1 = 2. When < 1, a higher increases

the coefcients on uture output and ination but also leads to the inclusion o a new termcomposed o uture dierences between output growth and interest rates. Each o these eects

makes price-setting decisions more orward-looking. The increased coefcient on expectations

o uture ination, which reects the expected uture depreciation o the reset price and the losses

associated with it, plays a particularly important role. In response to an inationary shock, a frm

1378
http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

9/37


that can reset its price will expect higher ination today and in the uture as other frms updatetheir prices in response to the shock. Given this expectation, the more orward looking a frm is

(the higher ), the greater the optimal reset price must be in anticipation o other frms raising

their prices in the uture. Thus, reset prices become more responsive to current shocks with

higher . We confrm numerically that this eect dominates the reduced sensitivity o ination to

the reset price in equation (21), thereby endogenously generating a positive relationship betweenthe level and the volatility o ination.

To close the model, we assume that the log-deviation o the desired gross interest rate rom

its steady-state value (rt ) ollows a Taylor rule

rt = 1rt1 + 2r

t2 + (1 1 2)

(t

) + y (yt y) + gy (gyt gy

)

+ p(pt pt )

+ rt ,

where , y , gy and p capture the strength o the policy response to deviations o ination,

the output gap, the output growth rate, and the price level rom their respective targets, parame-ters 1 and 2 reect interest rate smoothing, while

rt is a policy shock. We set

= , pt =

t = t,y = y, and gy = gy so that the central bank has no inationary or output bias. The

growth rate o output is related to the output gap by

gy t = yt yt1. (23)Since the actual level o the net interest rate is bounded by zero, the log deviation o the gross

interest rate is bounded by rt = log(Rt) logR

logR

= r with the dynamics o the

actual interest rate given by

rt = max{rt , r}. (24)

We consider the Taylor rule a reasonable benchmark because it is likely to be the closest

description o the current policy process and because suggestions to raise the optimal inationrate are not commonly associated with simultaneous changes in the way that stabilization policy

is conducted. However, in Section 6.1, we also derive the optimal given optimal stabilization

policy under discretion and commitment.

2.3. Shocks

We assume that technology ollows a random walk process with drit:

at = at1 + + at . (25)

Each o the risk premium, government, and Phillips Curve shocks ollow AR(1) processes

qt = q qt1 + qt , (26)

gt = g gt1 + g1y

gt , (27)

mt = m m t1 + mt . (28)

We assume that at , qt ,

gt ,

mt and

rt are mutually and serially uncorrelated.

1379
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

10/37


2.4. Welare unction

To quantiy welare or dierent levels o steady-state ination, we use a second-order approx-

imation to the household utility unction as in Woodord (2003).4 The main result can be sum-marized by the ollowing proposition, with all proos in online Appendix A.

Proposition 1. The second-order approximation to expected per period utility in equation (1)is5

0 + 1var( yt) + 2var(t) + 3var(ct), (29)

where parameters i , i = {0, 1, 2} depend on the steady-state ination and are given by

0 = 1 (1 )(1 gy ) (1 h)(1 h) (1 (1 + 1)Q0y) logX (1 )(1 + 1)

2(1 gy)

(1 h)

(1 h) logX2

(1 )

(1 gy )

(1 h)

(1 h)

(1 + 1)[Q0y ]

2 Q0y +2(1 + 1)

2

1 = (1 + 1)

2(1 gy)

(1 h)

(1 h),

2 = 2

2(1 gy)

(1 h)

(1 h)3

Q1y (1 1) + (1 + 1)1 + 1 Q0y Q1y (1 + 1) 1 Q1y logX ,3 =

h(1 c)

(1 h)2

4. In our welare calculations, we use the second-order approximation to the consumer utility unction while the

structural relationships in the economy are approximated to frst order. As discussed in Woodord (2010), this approach

is valid i distortions to the steady state are small so that the frst-order terms in the utility approximation are premulti-

plied by coefcients that can also be treated as frst-order terms. Since given our parameterization the distortions rom

imperect competition and ination are small (as in Woodord, 2003), this condition is satisfed in our analysis. Further-

more, we show in online Appendix F that the log-linear solution closely approximates the non-linear solution, which

implies that second-order eects on the moments o ination and output are small and can be ignored in the welare

calculations.

5. The complete approximation also contains three linear terms, the expected output gap, expected consumption,

and expected ination. Since the distortions to the steady state are small or the levels o trend ination we consider,

the coefcients that multiply these terms can be considered as frst order so we can evaluate these terms using the

frst-order approximation to the laws o motion as in Woodord (2003). We confrmed in numeric simulations that

they can be ignored. Furthermore, second-order eects on the expected output gap and expected ination are likely to

be quantitatively small since the linear solution closely approximates the non-linear solution to the model (see online

Appendix F).

1380

7/28/2019 82770969.pdf

11/37


0 = {1 + ( 1)Q1p[(1 )(b +Q

0p) (1 ) ]}

1,

1 = {1 ( 1)(1 ) Q1p}0,

3 =0

1 1{(1 )M2 + },

Q0y = 1

2

1 +

1

2

1

2

2,

Q1y =

1

1

2

1

2

1 +

1

2

1

2

3,

Q0p =1

2

1 +

1

2(1 )2

2,

Q1p = 1 12 (1 )21 + 12 (1 )23, = 2

(1 )2

(1 )2, = log

1

, and c = corr(ct, ct1).

This approximation o the household utility places no restrictions on the path o nominal

interest rates and thus is invariant to stabilization policies chosen by the central bank.

The loss unction in Proposition 1 illustrates the three mechanisms via which trend inationaects welare: the steady-state eects, the eects on the coefcients o the utility unction

approximation, and the dynamics o the economy via the second moments o macro-economic

variables.6 First, the term 0 captures the steady-state eects rom positive trend ination, which

hinge on the increase in the cross-sectional steady-state dispersion in prices (and thereore ininefcient allocations o resources across sectors) associated with positive trend ination.7 Note

that as approaches zero, 0 converges to zero. As shown by Ascari and Ropele (2007), when

= 0, 0/ > 0, but the sign o the slope quickly reverses at marginally positive ination

rates. In our baseline calibration, 0 is strictly negative and 0/ < 0 when trend ination

exceeds 012% per annum. Thus, or quantitatively relevant ination rates, the welare loss romsteady-state eects is increasing in the steady-state level o ination. This is intuitive since,

except or very small levels o ination, the steady-state level o output declines with higher because the steady-state cross-sectional price dispersion rises. The steady-state cost o ination

rom price dispersion is one o the best-known costs o ination and arises naturally rom the

integration o positive trend ination into the New Keynesian model. Consistent with this eect

being driven by the increase in dispersion, one can show that the steady-state eect is eliminatedwith ull indexation o prices and mitigated with partial indexation.

6. When = 0, equation (41) reduces to the standard second-order approximation o the utility unction as in

Proposition 6.4 oWoodord (2003). There is a slight dierence between our approximation and the approximation in

Woodord (2003) since we ocus on the per-period utility while Woodord calculated the present value.

7. The parameter measures the deviation o the exible-price level o output rom the exible-price perect-

competition level o output. See Woodord (2003) or derivation.

1381

7/28/2019 82770969.pdf

12/37


Second, the coefcient on the variance o output around its steady-state 1 < 0 does notdepend on trend ination. This term is directly related to the increasing disutility o labour

supply. With a convex cost o labour supply, the expected disutility rises with the variance o

output around its steady-state. However, even though 1 is independent o , this does not

imply that a positive does not impose any output cost. Rather, trend ination reduces the

steady-state level o output, which is already captured by 0. Once this is taken into account,then log utility implies that a given level o output variance around the (new) steady-state is as

costly as it was beore. Furthermore, the variance o output around its steady-state depends on

the dynamic properties o the model which are aected by the level o trend ination.

The coefcient on the variance o ination 2 < 0 captures the sensitivity o the welare loss

due to the cross-sectional dispersion o prices. One can also show analytically that or 0,

2/ < 0 so that the cross-sectional dispersion o prices becomes ceteris paribus costlier in

terms o welare. Because an inationary shock creates distortions in relative prices and positive

trend ination already generates some price dispersion and an inefcient allocation o resources,

frms operating at an inefcient level have to compensate workers or the increasingly high

marginal disutility o sector-specifc labour. With this rising marginal disutility, the increaseddistortion in relative prices due to an ination shock becomes costlier due to the higher initial

price dispersion making the variance o ination costlier or welare as the trend level o inationrises. This is a second, and to the best o our knowledge previously unidentifed, channel through

which the price dispersion rom staggered price setting under positive ination reduces welare.

Finally, the coefcient on the variance o consumption 3 < 0 captures the desire o habit-driven consumers to smooth consumption. The greater the degree o habit ormation, the more

costly a given amount o consumption volatility becomes. Conversely, the greater the autocor-

relation o consumption, the smaller are period-by-period changes in consumption, and the less

costly consumption volatility becomes. Trend ination changes this coefcient only by aecting

the persistence o consumption.

3. CALIBRATION AND OPTIMAL INFLATIONHaving derived the approximation to the utility unction, we now turn to solving or the optimal

ination rate. Because utility depends on the volatility o macro-economic variables, this will

be a unction o the structural parameters and shock processes. Thereore, we frst discuss our

parameter selection and then consider the implications or the optimal ination rate in the model.

3.1. Parameters

Our baseline parameter values are illustrated in Table 1. For the utility unction, we set , the

Frisch labour supply elasticity, equal to one. The steady-state discount actor is set to 0998

to match the real rate o 23% per year on 6-month commercial paper or assets with similar

short-term maturities given that we set the steady-state growth rate o real GDP per capita to be15% per year (GY = 1015025), as in Coibion and Gorodnichenko (2011). We set the elasticity

o substitution across intermediate goods to 7, so that steady-state markups are equal to 17%.

This size o the markup is consistent with estimates presented in Burnside (1996) and Basu

and Fernald (1997). The degree o price stickiness () is set to 055, which amounts to frms

resetting prices approximately every 7 months on average. This is midway between the microestimates o Bils and Klenow (2004), who fnd that frms change prices every 45 months, and

those oNakamura and Steinsson (2008), who fnd that frms change prices every 911 months.

The implied sensitivity o ination to marginal costs is 011, consistent with estimates rom

Altig et al. (2010).

1382
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

13/37


TABLE 1Baseline parameter values

Parameters o utility unction Steady-state values

: Frisch labour elasticity 100 gy : Growth rate o RGDP/cap 15%p.a.

: Discount actor 0998 cy : Consumption share o GDP 080h: Habit in consumption 07 gy : Government share o GDP 020

Pricing parameters Shock persistence

: Elasticity o substitution 7 g : Government spending shocks 097

: Degree o price stickiness 055 m : Cost-push shocks 090

: Price indexation 000 q : Risk premium shocks 0947

Taylor rule parameters Shock volatility

: Long-run response to ination 250 g : Government spending shocks 00052

gy : Long-run response to output growth 150 m : Cost-push shocks 00014

y : Long-run response to output gap 011 q : Risk premium shocks 00024

1: Interest smoothing 105 a : Technology shocks 00090

2: Interest smoothing 013 r: Monetary policy shocks 00024

Notes: The table presents the baseline parameter values assigned to the model in section 3.1 and used or solving or the

optimal ination rate in Section 3.2. p.a. means per annum.

The degree o price indexation is assumed to be zero in the baseline or three reasons. First,

the workhorse New Keynesian model is based only on price stickiness, making this the most

natural benchmark (Woodord, 2003). Second, any price indexation implies that frms are con-

stantly changing prices, a eature strongly at odds with the empirical fndings oBils and Klenow

(2004), and more recently Nakamura and Steinsson (2008), among many others. Third, whileindexation is oten included to replicate the apparent role or lagged ination in empirical es-

timates o the New Keynesian Phillips Curve (NKPC; see Gali and Gertler, 1999), Cogley andSbordone (2008) show that once one controls or steady-state ination, estimates o the NKPC

reject the presence o indexation in price-setting decisions. However, we relax the assumption

o no indexation in the robustness checks.The coefcients or the Taylor rule are taken rom Coibion and Gorodnichenko (2011). These

estimates point to strong long-run responses by the central bank to ination and output growth

(25 and 15, respectively) and a moderate response to the output gap (043).8 The steady-state

share o consumption is set to 080 so that the share o government spending is 20%. The calibra-

tion o the shocks is primarily taken rom the estimated dynamic stochastic general equilibriummodel o Smets and Wouters (2007) with the exception o the persistence o the risk premium

shocks or which we consider a larger value calibrated at 0947 to match the historical requency

o hitting the ZLB and the routinely high persistence o risk premia in fnancial time series.9

In our baseline model, positive trend ination is costly because it leads to more price dis-

persion and thereore less efcient allocations, more volatile ination, and a greater welarecost or a given amount o ination volatility. On the other hand, positive trend ination gives

8. Because empirical Taylor rules are estimated using annualized rates while the Taylor rule in the model is

expressed at quarterly rates, we rescale the coefcient on the output gap in the model such that y = 043/4 = 011.

9. This calibration is, or example, consistent with the persistence o the spread between Baa and Aaa bonds

which we estimate to be 0945 between 1920:1 and 2009:2 and 0941 between 1950:1 and 2009:2 at the quarterly

requency.

1383
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

14/37


FIGURE 1

Frequency o being in the ZLB and steady-state nominal interest rate. Notes: The fgure plots the steady-state level o

the annualized nominal interest rate (right axis) implied by the baseline model o Section 3 or dierent steady-state

ination rates. In addition, the fgure plots the requency o hitting the zero bound on nominal interest rates (let axis)

rom simulating the baseline model at dierent steady-state ination rates as well as historical requencies o ZLB or

the U.S. or counteractual ination rates (see Section 3.1 or details)

policymakers more room to avoid the ZLB on interest rates. Thereore, a key determinant o the

trade-o between the two depends on how requently the ZLB is binding or dierent levels o

trend ination. To illustrate the implications o our parameter calibration or how oten we hitthe ZLB, Figure 1 plots the raction o time spent at the ZLB rom simulating our model ordierent steady-state levels o the ination rate. In addition, we plot the steady-state level o the

nominal interest rate associated with each ination rate, where the steady-state nominal rate in

the model is determined by R = .GY/. Our calibration implies that with a steady-state ina-

tion rate o35%, the average rate or the U.S. since the early 1950s, the economy should be

at the ZLB 5% o the time. This is consistent with the post-WWII experience o the U.S.: withU.S. interest rates at the ZLB since late 2008 and expected to remain so by the end o 2011, this

yields a historical requency o being at the ZLB o 5% (i.e. around 3 years out o 60), although

we also consider much higher requencies in Section 4.2.10 For example, we show that our

results are qualitatively robust to assuming that the current ZLB episode can last until 2017

(i.e. a ZLB requency o 15% at 3% trend ination), thereby corresponding to a ull lost decadear exceeding in length the current Fed commitment to sustain exceptionally low interest rates

until at least mid-2013.

In addition, our baseline calibration agrees with the historical changes in interest rates as-

sociated with post-WWII U.S. recessions. For example, starting with the 1958 recession and

excluding the current recession, the average decline in the Federal Funds Rate (FFR) during a

10. O possible concern may be that this calculation includes the high-ination environment rom 1970 to 1985.

Excluding those years generates a historical requency at the ZLB o 3/45 = 6 66% but now at a lower trend ination

rate o 3% per year. Our baseline calibration generates approximately that requency at 3% trend ination.

1384
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

15/37


recession has been 476 percentage points.11 The model predicts that the average nominal inter-est rate with 35% steady-state ination is around 6%, so the ZLB would not have been binding

during the average recession, consistent with the historical experience. Only the 19811982

recession led to a decline in nominal interest rates that would have been sufciently large to

reach the ZLB (866% drop in interest rates) but did not because nominal interest rates and es-

timates o trend ination over this period were much higher than their average values. Thus,with 335% ination, our calibration (dotted line in Figure 1) implies that it would take unusu-

ally large recessions or the ZLB to become binding. In addition, our calibration indicates that

at much lower levels o , the ZLB would be binding much more requently: or example, at

= 0, the ZLB would be binding 27% o the time. This seems conservative since it exceeds the

historical requency o U.S. recessions. The model predicts a steady-state level o interest rateso

7/28/2019 82770969.pdf

16/37


FIGURE 2

Utility at dierent levels o steady-state ination Panel A: Eect o ZLB, Panel B: Eects o positive trend ination.

Notes: The fgures plot the approximation to the utility unction in Proposition 1 rom simulating the model or dierent

levels o steady-state ination. Panel A includes results or the baseline model, the baseline model without the ZLB,as well as the model with the ZLB but omitting the three cost channels o ination: steady-state eects, the changing

coefcient on ination variance in utility, and the dynamic eects. Panel B reproduces our baseline with ZLB then

presents results when we restrict the model to include only one cost o ination and the ZLB. Dynamic cost only

includes only the dynamic eects o positive ination and keeps the rest o the model being approximated around zero

trend ination, Steady-state cost only includes only the steady-state cost o ination and keeps the rest o the model

being approximated around zero trend ination, while Changing ination weight only includes only the changing

coefcients on ination variance in the loss unction and keeps the rest o the model being approximated around zero

trend ination (see Section 3.2 or details)

that this algorithm has very high accuracy even ater large shocks leading to a binding ZLB.

The results taking into account the ZLB and in the case when we ignore the ZLB are plottedin Panel A o Figure 2. When the ZLB is not taken into account, the optimal rate o ination iszero because there are only costs to ination and no benefts. Figure 2 also plots the other ex-

treme when we include the ZLB but do not take into account the eects o positive steady-state

ination on the loss unction or the dynamics o the model. In this case, there are no costs to in-

ation so utility is strictly increasing as steady-state ination rises and the requency o the ZLB

diminishes. Our key result is the specifcation which incorporates both the costs and benefts oination. As a result o the ZLB constraint, we fnd that utility is increasing at very low levels o

ination so that zero ination is not optimal when the zero bound is present. Second, the peak

level o utility is reached when the ination rate is 1 5% at an annualized rate. This magnitude is

close to the bottom end o the target range o most central banks, which are commonly between

1% and 3%. Thus, our baseline results imply that taking into account the zero bound on interestrates raises the optimal level o ination, but with no additional benefts to ination included in

the model, the optimal ination rate is within the standard range o ination targets. Third, the

costs o even moderate ination can be non-trivial: a 5% ination rate would lower utility by

1% relative to the optimal level, which given log utility in consumption is equivalent to a per-

manent 1% decrease in the level o consumption. As we show later, the magnitude o the welarecosts o ination varies with the calibration and price-setting assumptions, but the optimal rate

o ination is remarkably insensitive to these modifcations.

Panel B o Figure 2 quantifes the importance o each o the three costs o inationthe

steady-state eect, the increasing cost o ination volatility, and the positive link between the

1386

7/28/2019 82770969.pdf

17/37


level and volatility o inationby calculating the optimal ination rate subject to the ZLB whenonly one o these costs, in turn, is included. The frst fnding to note is that allowing or any o

the three ination costs is sufcient to bring the optimal ination rate to 36% or below. Thus,

all three ination costs incorporated in the model are individually large enough to prevent the

ZLB rom pushing the optimal ination rate much above the current target range o most central

banks. Second, the steady-state cost is the largest cost o ination out o the three, bringing theoptimal ination rate down to 16% by itsel. However, even i we omit steady-state costs and

include only the other two channels, the optimal ination rate would be

7/28/2019 82770969.pdf

18/37


FIGURE

3

Thesourceso

utilitycostso

in

ation.

Notes:The

frstrowo

the

fgurep

lotsthecoe

fc

ientso

theapprox

imation

totheutilityunction

rom

Proposition

1

or

dierentleve

lso

tren

dination.

Thesecon

drowplotsthevarianceo

macro-econom

icvariab

lesthatentertheapprox

imationtotheuti

lityunction

inProposition

1romsimulat

ingthemo

de

lsu

bject

tothezero

boun

donnom

ina

linterestrates

or

dierentleve

lso

stea

dy-s

tate

inationusingthe

base

lineparameterva

lueso

themo

de

l.The

dashe

dblac

klinesarethecorrespon

ding

momentsw

ithoutthezero

boun

do

nnom

ina

linterestrates,w

hilethe

dotted

linescorrespon

dtothemomentso

themo

de

lw

henthe

dynam

icsareapprox

imatedatzerotren

dination.

Thethirdrowp

lots

thecontributiono

the

dierentcomponentso

theapprox

imationtotheutilityunc

tion

inProposition

1(see

Section

3.2

ord

etails)

1388

7/28/2019 82770969.pdf

19/37


FIGURE 4

The costs o business cycles. Panel A: The costs o the ZLB and Panel B: Changing the weight on output gap volatility

in the loss unction. Notes: The top panel plots the average duration o ZLB episodes in the baseline calibration and the

implied average welare cost per quarter o being at the ZLB or dierent levels o trend ination. The bottom panel

plots the eects o changing the coefcient on the variance o the output gap in the utility unction approximation o

Proposition 1 on the optimal ination rate (let graph), the welare costs o business cycle uctuations (middle graph),

and the average welare costs o hitting the ZLB (right graph). The latter two are measured using Proposition 1 net o

the steady-state eects o trend ination (see Sections 3.3 and 3.4 or details)

3.3. Are the costs o business cycles and the ZLB too small in the model?

The minor contribution o output gap volatility to the optimal ination rate might be interpreted

as an indication that the model understates the costs o business cycles in general and the ZLB in

particular. For the ormer, the implied welare costs o business cycles in our model are 05% osteady-state consumption at the historical trend ination rate, in line with many o the estimates

surveyed in Barlevy (2004) and much larger than in Lucas (1987). To assess the cost o hitting the

ZLB, we compute the average welare loss net o steady-state eects rom simulating the model

under dierent ination rates both with and without the zero bound. The dierence between the

two provides a measure o the additional welare cost o business cycles due to the presenceo the ZLB. We can then divide this cost by the average requency o being at the zero bound

rom our simulations, or each level o steady-state ination, to get a per-quarter average welare

loss measure conditional on being at the ZLB which is plotted in Panel A o Figure 4. As rises, this per-period cost declines because the average duration o ZLB episodes gets shorter

and the output losses during the ZLB are increasing non-linearly with the duration o the ZLB(see Christiano, Eichenbaum and Rebelo, 2011). For example, the average cost o a quarter

spent at the ZLB is approximately equivalent to a permanent 14% reduction in consumption

when ination is 1% but declines to 04% at a 35% rate o ination. The latter implies that

the additional cost o being restrained by the zero bound or 8 quarters is equivalent to a 32%

1389

7/28/2019 82770969.pdf

20/37


permanentreduction in consumption or $320 billion per year based on 2008 consumption data.For comparison, Williams (2009) uses the Federal Reserves FRB/U.S. model to estimate that

the ZLB between 2009 and 2010 cost $18 trillion in lost output over 4 years or roughly $300

billion per year in lost consumption over 4 years i one assumes that the decline in consumption

was proportional to the decline in output. Thus, the costs o both business cycles and the ZLB in

the model cannot be described as being uncharacteristically small.However, while the conditional costs o long ZLB events are quite large, they also occur

relatively inrequently. For example, i we assume that all ZLB episodes are 8 quarters long,

then at 35% trend ination, an 8-quarter episode at the ZLB occurs with probability 0007 each

quarter or about 3 times every 100 years. This implies that the expected cost o the ZLB is a

002% permanent reduction o consumption. Similar calculations or 2% trend ination revealthat while the conditional cost o an 8-quarter ZLB event is about a 6 2% permanent reduction

o consumption, the unconditional cost o the ZLB is only a 008% permanent reduction in

consumption. Thus, while the model implies that a higher ination target can signifcantly reduce

the cost o a given ZLB event, as suggested by Blanchard, taken over a long horizon, the expected

gain in mitigating the ZLB rom such a policy is small. As a result, even modest steady-statecosts o ination, because they must be borne every period, are sufcient to push the optimal

ination rate below 2%.

3.4. How does optimal ination depend on the coefcient on the variance o the output gap?

Even though the costs o business cycles are signifcant and ZLB episodes are both very costly

and occurring with reasonable probability, one may be concerned that these costs are incor-

rectly measured due to the small relative weight assigned to output gap uctuations in the utility

unction. At = 0, the coefcient on the output gap variance in the loss unction is less thanone-hundredth that on the quarterly ination variance (or one-tenth or the annualized ina-

tion variance), and this dierence becomes even more pronounced as rises. The low weight

on output gap volatility is standard in New Keynesian models and could reect the lack o

involuntary unemployment, which inicts substantial hardship to a raction o the populationand whose welare eects may be poorly approximated by changes in aggregate consumptionand employment, or the absence o distributional eects, as business cycles disproportionately

impact low income/wealth agents with higher marginal utilities o consumption than the average

consumer.

To assess how sensitive the optimal ination rate is to the coefcient on the output gap vari-

ance, we increase this coefcient by a actor ranging rom 1 to 100 and reproduce our results orthe optimal or each actor (see Panel B o Figure 4). Raising the coefcient on the variance

o the output gap pushes the optimal ination rate higher, but the coefcient on the output gap

variance needs to be very large to qualitatively aect our fndings. For example, much o the

traditional literature on optimal monetary policy assumed equal weights on output and annu-

alized ination variances in the loss unction. With ination being measured at an annualizedrate, this equal weighting obtains at zero steady-state ination when 1 is multiplied by a actor

o10. Yet this weighting would push the optimal ination rate up only modestly to 1 6% per

year. Even i one increased the coefcient on output gap volatility by a actor o 100, the optimal

ination rate would rise only to 24%. Placing such weight on output volatility would raise the

implied per-quarter cost o having the ZLB bind to an equivalent o a 3% permanent reductionin consumption, such that an episode o 8 consecutive quarters at the ZLB would deliver welare

losses equivalent to roughly 24% o steady-state consumption, above and beyond the costs o

the shock in the absence o the ZLB. Thus, while one can mechanically raise the optimal ina-

tion rate via larger weights on output uctuations than implied by the model, weighting schemes

1390

7/28/2019 82770969.pdf

21/37


FIGURE 5

Sensitivity: pricing parameters, habit, and labour supply elasticity. Notes: Figures plot the optimal level o steady-state

ination and the welare loss at the optimal steady-state level o ination as a unction o a structural parameter (see

Section 4.1 or details)

that meaningully raise the optimal ination rate point to welare costs o business cycles, andparticularly episodes at the ZLB, that depart rom the conventional wisdom.

4. ROBUSTNESS OF THE OPTIMAL INFLATION RATE TO ALTERNATIVE

PARAMETER VALUES

In this section, we investigate the robustness o the optimal ination rate to our parameterizationo the model. We ocus particularly on pricing and utility parameters, the discount actor, and

the risk premium shock.

4.1. Pricing and utility parameters

Figure 5 plots the optimal ination rates and associated welare losses or dierent levels o

or alternative pricing and utility parameters. First, we consider the role o the elasticity o

substitution (Panel A), which plays a critical role in determining the cost o price dispersion,

and thereore costs o ination, in the model. Note that the welare costs o ination are larger

when is high. This result captures the act that a higher elasticity o substitution generatesmore steady-state output dispersion and, through greater real rigidity, higher welare cost o

uctuations or any . However, the eects o this parameter on the optimal are relatively

small: even a value o = 3 yields an optimal o

7/28/2019 82770969.pdf

22/37


the amount o price and output dispersion and thereore the steady-state costs o ination. Thiseect would likely be particularly pronounced i price changes within industries were synchro-

nized so that the relevant amount o price dispersion or welare arises primarily rom cross-

industry substitution at low elasticities.

We can assess the implications o these possibilities by introducing a ew modifcations to

our baseline model. Specifcally, we can consider a two-tier model in which the elasticity osubstitution within industries is given by and the elasticity o substitution across industries is

given by , where we assume > . Prices are sticky at the frm level but not at the indus-

try levelindustries simply combine individual goods and sell them o to consumers. In the

absence o industry-specifc shocks and price synchronization within industries, average price

levels across industries must be equal since industries are symmetric and Calvo price stickinesswashes out at the industry level. As a result, there would be no price dispersion across industries.

In addition, i we assume that there is a continuum o frms that aggregate or a given industry i ,

then there is no industry markup. Thus, the steady-state markup on goods would be 1 and this

model would have exactly the same implications as our baseline model with all the results being

driven by rather than . This suggests that, absent price synchronization, it is the elasticity osubstitution within industries, which is most relevant or our results. Intuitively, since all indus-

tries have the same price index and the same intra-industry price dispersion, we can combinethem into a representative industry, thereby reducing this model to our baseline setup.

An important potential caveat to this result, however, is i price changes are synchronized

within industries, as in Bhaskar (2002). Price synchronization within industries could signi-icantly reduce price dispersion within industries while the remaining price dispersion across

industries would matter little or welare under low cross-industry elasticities o substitution.

To quantiy this possibility, we used a model with Taylor pricing (as in Section 5.4) with our

industries, each o which had staggered pricing over our quarters but in which disproportionate

shares o frms within each industry reset their prices in the same quarter. We allowed or theelasticity o substitution within industries ( = 7) to exceed that across industries ( = 3). Even

high levels o synchronization (i.e. 70% o frms in an industry resetting their prices in the same

quarter) had only modest eects on steady-state levels o price dispersion and thus on our wel-are calculations.12 While these modifcations suggest that our results are robust to a number o

alternative assumptions about industrial structure and pricing assumptions, one should nonethe-less bear in mind that there is signifcant uncertainty about how best to model the substitution and

pricing o goods across industries and that alternative specifcations may quantitatively aect the

predictions about the costs o price dispersion.

Second, we also investigate the role o price indexation. In our baseline, we assumed = 0,

based on the act that frms do not change prices every period in the data, as documented by Bilsand Klenow (2004) and Nakamura and Steinsson (2008), as well as the results o Cogley and

Sbordone (2008) who argue that once one controls or time-varying trend ination, we cannot

reject the null that = 0 or the U.S. However, because price indexation is such a common

component o New Keynesian models, we consider the eects o price indexation on our re-sults. Panel B o Figure 5 indicates that higher levels o indexation lead to higher optimal rates

12. These results are available upon request. The empirical evidence on price synchronization is mixed. Dhyne and

Konieczny (2007), or example, document that price synchronization within industries is larger than across industries but

also document a remarkable degree o staggering o price changes within industries. For example, they observe (p. 11),

. . . price changes at the individual product category level are neither perectly staggered nor synchronized, but their

behavior is much closer to perect staggering. Likewise, a more recent study, Klenow and Malin (2011) use data rom

U.S. Bureau o Labor Statistics and fnd that the timing o price changes is little synchronized across sellers even within

very narrow product categories.

1392
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

23/37


FIGURE 6

Sensitivity: risk premium shocks and the discount actor. Notes: Figures in the let column plot the welare loss as a

unction o steady-state ination or alternative values o structural parameters. The solid thick back line corresponds to

the baseline parameterization. Figures in the middle column plot the requency o hitting the ZLB or dierent parameter

values. Figures in the right column plot the optimal level o steady-state ination and the welare loss at the optimal

steady-state level o ination as a unction o a structural parameter (see Section 4.2 or details)

o ination because indexation reduces the dispersion o prices. Yet with = 0 5, which is

most likely an upper bound or an empirically plausible degree o indexation in low-ination

economies like the U.S., the optimal remains

7/28/2019 82770969.pdf

24/37


dierent levels o the persistence to risk premium shocks. The results are quite sensitive to thisparameter, which reects the act that these shocks play a crucial role in hitting the ZLB. For

example, Figure 6 illustrates that when we raise the persistence o the shock rom 0947 to 096,

the optimal ination rate rises rom 15% to 3% because this increase in the persistence o the

shock has a large eect on the requency and duration o being at the ZLB. At 35% ination,

this requency more than doubles relative to our baseline scenario, thereby raising the beneft ohigher steady-state ination. The reverse occurs with lower persistence o risk premium shocks:

the requency o being at the ZLB declines sharply as does the optimal ination rate. Second,

similar results obtain when we vary the volatility o the risk premium shock. When we increase

the standard deviation o these shocks to 00035 rom our baseline o 00024, the optimal ina-

tion rate again rises to slightly over 3%. As with the persistence o the shocks, this is driven by ahigher requency o being at the ZLB: at 35% ination, this alternative shock volatility implies

the economy would be at the ZLB three times as oten as under our baseline calibration.

Third, we consider the sensitivity o our results to the steady-state level o the discount actor

. This parameter is also important in determining the requency at which the economy is at the

ZLB since it aects the steady-state level o nominal interest rates. As with the risk premiumshock variables, a higher value o is associated with a lower steady-state level o nominal

interest rates, so that the ZLB will be binding more requently. For example, with = 09999(which corresponds to a real rate o 154% per year), the ZLB is binding 7% o the time when

steady-state ination is 35%. At the maximum, however, the optimal is only 06% higher than

implied by our baseline results.These robustness checks clearly illustrate how important the requency at which the economy

hits the ZLB is or our results. Naturally, parameter changes that make the ZLB binding more

oten raise the optimal rate o ination because a higher lowers the requency o hitting the

ZLB. Thus, the key point is not the specifc values chosen or these parameters but rather hav-

ing a combination o them that closely reproduces the historical requency o hitting the ZLBor the U.S. Nonetheless, even i we consider parameter values that double or even triple the

requency o hitting the ZLB at the historical average rate o ination or the U.S., the optimal

ination rate rises only to about 3%. This suggests that the evidence or an ination target in theneighbourhood o 2% is robust to a wide range o plausible calibrations o hitting the ZLB.

4.3. Summary

These results indicate that the optimal ination rate in the baseline model is robust to reasonable

variations in the parameters as calibrated to the U.S. economy. However, the act that more

persistent and volatile shocks could potentially raise the requency o the ZLB suggests that the

optimal ination rate is likely to vary across countries. For example, smaller open economies

are typically more subject to volatile terms o trade shocks than the U.S. and this increasedvolatility could justiy higher target rates o ination i it increases the incidence o hitting the

ZLB. Similarly, economies such as Japan in which the real interest rate has historically been

very low might also fnd it optimal to pursue higher ination targets. More broadly, our results

highlight the importance o careully calibrating the model parameters to the specifcs o the

economy beore drawing general conclusions about optimal policies.

5. WHAT COULD RAISE THE OPTIMAL INFLATION RATE

While our baseline model emphasizes the trade-o between higher to insure against the zero

bound on nominal interest rates versus the utility costs associated with higher trend ination,

previous research has identifed additional actors beyond the lower bound on nominal interest

1394
http://-/?-http://-/?-

7/28/2019 82770969.pdf

25/37


rates that might lead to higher levels o optimal ination. In this section, we extend our analysisto assess their quantitative importance. First, we include capital ormation in the model. Second,

we allow or uncertainty about parameter values on the part o policymakers. Third, we inte-

grate downward nominal wage rigidity, that is, greasing the wheels into the model. Fourth,

we explore whether our results are sensitive to using Taylor pricing. Finally, we consider the

possibility that the degree o price stickiness varies with .

5.1. Capital

First, we consider how sensitive our results are to the introduction o capital. We present a

detailed model in online Appendix B and only provide a verbal description in this section. Inthis model, frms produce output with a CobbDouglas technology (capital share = 033).

All capital goods are homogeneous and can be equally well employed by all frms. Capital is

accumulated by the representative consumer subject to a quadratic adjustment cost to capital

( = 3 as in Woodord, 2003) and rented out in a perectly competitive rental market. Theaggregate capital stock depreciates at rate = 002 per quarter. We calculate the new steady-state

level o output relative to the exible-price level output and derive the analogue o Proposition

1 in online Appendix B with proos in online Appendix C.By allowing capital to reely move between frms, we reduce the steady-state welare cost

rom trend ination since frms that have a relatively low price can now hire additional capitalrather than sector-specifc workers to boost their output. Capital also increases the likelihood

o hitting the ZLB and thereore the benefts o ination because including capital permits dis-

invesment when agents preer storing wealth in sae bonds rather than capital, so we are more

likely to be in a situation where an increase in qt pushes interest rates to zero. We isolate the

frst channel by setting q = 0943 to match the historical requency at the ZLB. As shown in

Panel A o Figure 7, utility peaks at a trend ination rate o 21% suggesting that capital doesnot lower the cost o ination substantially.

5.2. Model uncertainty

An additional eature that could potentially lead to higher rates o optimal ination is uncer-

tainty about the model on the part o policymakers. I some plausible parameter values lead to

much higher requencies o hitting the ZLB or raise the output costs o being at the ZLB, then

policymakers might want to insure against these outcomes by allowing or a higher . We quan-

tiy this uncertainty via the variancecovariance matrix o the estimated parameters rom Smetsand Wouters (2007), placing an upper bound on parameter values to eliminate draws where

the ZLB binds unrealistically oten in excess o 10% at the historical average o annual trend

ination.

To assess the optimal ination rate given uncertainty about parameter values, we compute the

expected utility associated with each level o steady-state ination by repeatedly drawing romthe distribution o parameter values. Panel B o Figure 7 plots the implied levels o expected

utility associated with each steady-state level o ination. Maximum utility is achieved with

an ination rate o 19% per year. As expected, this is higher than our baseline result, which

reects the act that some parameter draws lead to much larger costs o being at the zero bound,

a eature that also leads to a much more pronounced inverted U-shape o the welare losses romsteady-state ination. Nonetheless, this optimal rate o ination remains well within the bounds

o current ination targets o modern central banks.

We also consider another exercise in which we repetitively draw rom the parameter space

and solve or the optimal ination rate associated with each draw. The distribution o optimal

1395
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

26/37


ination rates has a 90% confdence interval ranging rom 03% to 29% per year, which againis very close to the target range or ination o most central banks.

5.3. Downward wage rigidity

A common motivation or positive trend ination, aside rom the ZLB, is the greasing the wheels

eect raised by Tobin (1972). I wages are downwardly rigid, as usually ound in the data (e.g.Dickens et al. 2007), then positive trend ination will acilitate the downward adjustment o

real wages required to adjust to negative shocks. To quantiy the eects o downward nominal

wage rigidity in our model, we integrate it in a manner analogous to the zero bound on interestrates by imposing that changes in the aggregate nominal wage index be above a minimum bound

wt = max{wmt , w

t }, where w

t is the change in wages that would occur in the absence

o the zero bound on nominal wages and wmt is the lower bound on nominal wage changes.

Note that even with zero steady-state ination, steady-state nominal wages grow at the rate o

technological progress. Thus, we set wmt to be equal to minus the sum o the growth rate otechnology and the steady-state rate o ination.

Panels C and D o Figure 7 present the utility associated with dierent under both the

zero bound on interest rates and the downward wage rigidity. The result is striking: the optimalination rate alls to 03% per year with downward wage rigidity. With downward wage rigidity,

marginal costs are less volatile, so the variance o ination is reduced relative to the case withexible wages. In addition, the act that marginal costs are downwardly rigid means that, in the

ace o a negative demand shock, ination will decline by less and thereore interest rates will

all less, reducing the requency o the ZLB. With = 0, the ZLB binds

7/28/2019 82770969.pdf

27/37


FIGURE 7

Robustness. Notes: Panel A plots welare in the baseline model and the model with capital (see Section 5.1 or details).

Panel B plots the expected utility in the baseline model and model uncertainty (see Section 5.2 or details). Panel C

plots the utility associated with dierent steady-state ination rates under the baseline model as well as the model with

downward nominal wage rigidity. Panel D plots the variance o ination or dierent steady-state ination rates using

our baseline model and the model with downward nominal wage rigidity (see Section 5.3 or details). Panels E and F

plot the implications o Calvo vs. Taylor price setting or welare. Taylor, X quarters corresponds to the duration o price

contracts equal to X quarters (see Section 5.4 or details)

1397
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

28/37


Coibion, Gorodnichenko and Wieland, 2011 or more details) are approximately the same in allmodels. Instead, the main dierence is that the Taylor model has smaller welare losses as increases above 2% because o the reduced steady-state price dispersion. Intuitively, since frms

under Calvo pricing may be stuck with a suboptimal price or a long time, the cost o positive

steady-state ination is larger than in the Taylor model where frms are guaranteed to change

prices in a fxed number o periods. The quantitative implications or the optimal ination rate,however, are quite small.

5.5. Endogenous and state-dependent price stickiness

Because theory implies that the cost to frms o not changing prices should increase as rises

(Romer, 1990), higher levels o ination should be associated with lower levels o price sticki-

ness, which would tend to lower the welare costs o positive trend ination. Thus, by ignoring

this endogeneity, we might be overstating the costs o positive ination and thereby underesti-

mating the optimal rate o ination. To address this possibility, we consider the sensitivity oour baseline results to a possible systematic link between the ination rate and the degree o

price stickiness in two ways. As a frst approach, we ollow Nakamura and Steinssons empirical

approach and posit a linear relationship between the (monthly) requency o price changes andthe steady-state annual rate o ination, with the coefcient on ination denoted by . The

average estimate o Nakamura and Steinsson across price measures and time periods is approx-imately = 05, and the upper bound o their confdence intervals is approximately = 1.

We calibrate the degree o price rigidity such that = 055 (our baseline value) at a steady-state

level o annual ination o 35%. In this setting, the degree o price stickiness varies with the

steady-state rate o ination but not with business cycle conditions.

As a second approach, we consider state-dependent pricing in the spirit o Dotsey, Kingand Wolman (1999). Using the same model as in Section 2, we replace the exogenous prob-

ability o changing prices with an explicit optimizing decision based on comparison o menu

costs and the gains rom price adjustment. Specifcally, we assume that each period, frms draw

rom a uniorm distribution o costs to changing prices and, conditional on their draw, decide

whether or not to reset their price. Because all frms are identical, every frm that choosesto reset its price picks the same price. As in the Taylor (1977) staggered contracts model,

the distribution o prices in the economy depends only on past reset prices and the share o

frms which changed their price in previous quarters, but, in contrast to time-dependent mod-

els, these shares are time-varying. Derivations o the model and the associated welare loss

unction are provided in online Appendix E. Using the same parameter values as the baselinemodel, we calibrate the average size o menu costs to yield the same degree o price stickiness

at a steady-state ination rate o 35% as in the baseline case. Because this calibration implies

strong strategic complementarity in price setting, large menu costs (7% o output) are nec-

essary to match the duration o price spells in the data.15 We thereore quantiy the sensitivity

o our results to this parameterization by varying the elasticity o substitution across interme-

diate goods rom = 10 to = 5 while simultaneously varying the menu costs rom 11% to4% o revenue to maintain the same average price duration at 35% ination as in our baseline

calibration.

Panels A and D in Figure 8 illustrate the implied variation in the degree o price stickiness or

both methods. In each case, higher steady-state levels o ination lead to more requent updating

15. The size o the menu costs is in line with those reported in Zbaracki et al. (2004) o6% o operating expenses.

However, Zbaracki et al. (2004) emphasize that these costs appear to be largely associated with multiproduct pricing

decisions, whereas in our model, frms produce a single good. Unortunately, there is little other direct empirical evidence

on the size o menu costs.

1398
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

29/37


FIGURE 8

Endogenous and state-dependent price stickiness Endogenous calvo parameter. State-dependent pricing (Dotsey, King

and Wolman, 1999). Notes: The fgures plot the implications o endogenous and state-dependent price stickiness on the

model. is the eect o steady-state ination on the requency o price changes 1 . = 0 is our baseline case o

exogenous price stickiness (see Section 5.5 or details)

o prices. Panels B and E plot the welare associated with dierent and dierent parameters

or each approach. In the frst approach, more endogeneity in the degree o price stickiness leads

to slightly lower optimal levels o ination. The act that the optimal ination rate declines when

1399
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

30/37


the sensitivity o price stickiness to ination rises may seem counterintuitive: higher inationshould lead to aster updating o prices and thereore less price dispersion. But the endogenous

rate o price stickiness also implies that ination volatility rises more rapidly with average in-

ation than in the baseline case which raises the cost o ination. Furthermore, a higher degree

o price stickiness at low ination rates reduces the severity o deationary spirals and thus the

benefts o higher trend ination. Given our parameter values and the range o ination ratesthat we consider, the latter two eects approximately oset the ormer at low levels o ina-

tion so endogenous price stickiness actually leads to slightly lower optimal ination rates than

the standard Calvo model. Second, the welare costs o ination at the optimal rate are rising

with endogenous price setting. This reects the act that, given the same low optimal rate o

ination, more endogeneity is associated with higher degrees o price stickiness and thereorea higher cost o ination when < 3.5%. Third, despite the act that the optimal rate o ina-

tion varies little with endogenous price stickiness, the costs o much higher are signifcantly

lower relative to our baseline because higher ination leads to more requent price changes and

thereore price dispersion rises less rapidly with steady-state ination than under constant price

stickiness.Similarly, allowing or state-dependent pricing lowers the optimal ination rate relative to our

baseline model as well as the welare losses rom higher ination rates, with the latter reectingthe reduced sensitivity o price dispersion to ination due to endogenous pricing decisions, as in

Burstein and Hellwig (2008).16 Reducing the size o menu costs raises the optimal ination rate,

but the eects are quantitatively small. In short, the state-dependent model confrms the resultsreached using the endogenous Calvo price durations: allowing or price stickiness to all with

ination lowers the optimal ination rate relative our baseline model, even as the costs o higher

ination are substantially reduced.

6. NORMATIVE IMPLICATIONS

We have so ar been treating the question o the optimal ination rate independently o the

systematic response o the central bank to macro-economic uctuations. In this section, we in-vestigate the implications o optimal stabilization policy, both under commitment and discretion,

or the optimal rate o ination. We then consider whether altering the parameters o the Taylorrule can lead to outcomes that approach those achieved using optimal stabilization policy with

commitment.

6.1. Optimal stabilization policy

In the baseline model, stabilization policy ollows a Taylor rule calibrated to match the historical

behaviour o the Federal Reserve. Following Giannoni and Woodord (2010) and Woodord

(2010), we derive the optimal policy rules under commitment and discretion and simulate the

model with our baseline parameter values.17 Panel A o Figure 9 plots expected utility under

both policies. When the central bank can commit to a particular policy rule, then the optimalination rate is practically zero (02%). This is not because the ZLB binds less requentlyit

actually binds much more requently at the optimal ination ratebut rather because the cost

o the ZLB with commitment is negligible. This reects the act that in the event o a large

shock, the central bank will promise to keep interest rates low or an extended period, which

16. For easier comparison with the other models, we present welare results net o menu costs. Including the menu

costs pushes the optimal ination rate down by 0 2%05% per year.

17. We assume that the central bank can always commit to a long-run ination target even i it cannot commit to a

particular stabilization policy rule. Thereore, there is no ination bias under this policy.

1400
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

7/28/2019 82770969.pdf

31/37


FIGURE 9

Positive implications: optimal policy. Notes: Panels A and B plot outcomes or three scenarios: optimal policy with

commitment (Ramsey); optimal policy without commitment (discretion); the baseline Taylor rule described in Section

3.1. Optimal policies with and without discretion are described in Section 6.1. Panels CF plot the welare loss as a

unction o or alternative values o the monetary policy rule parameters. The solid thick blue line corresponds to the

baseline parameterization. is the long-run response o interest rates to ination, g is the response to output growth,

y is the response to the output gap, and p is the response to the price-level gap (see Section 6.2 or details)

1401

7/28/2019 82770969.pdf

32/37


signifcantly reduces the impact o the shock and thus the cost o the ZLB. Under discretionarypolicy, the reverse is true. The inability o the central bank to promise to keep interest rates low

ater the ZLB constraint ceases to bind yields very large costs o the ZLB. As a result, optimal

policy entails a higher level o trend ination, reaching 27% in our baseline calibration, which

is still within the target range o most central banks today. 18 Thus, a positive but low optimal

ination rate does not hinge on the assumption that the central bank ollows a Taylor rule but alsoobtains with optimal policy under discretion. The commitment case suggests that an improved

monetary policy could deliver important welare gains by reducing the costs o the ZLB when

they occur rather than trying to avoid these episodes.

6.2. Taylor rule parameters, PLT and the optimal ination rate

Given the size o the welare dierential between optimal policy with discretion/Taylor rule

versus commitment, we investigate to what extent alternative monetary policy rules can improve

outcomes in the ace o the ZLB. Thus, we consider the implications o alternative parametervalues in the Taylor rule, illustrated in Panels CF o Figure 9. Varying the long-run response

to ination rom 2 to 5 has unambiguous eects on welare: stronger long-run responses to

ination raise welare in the model or all ination rates. Intuitively, this stronger systematicresponse reduces ination and output volatility, thereby leading to a lower requency o being

at the ZLB and higher utility. However, this has little eect on the optimal ination rate, whichranges rom 17% when = 2 to 12% when = 5. In terms o responses to the real side

o the economy which are captured by coefcients y and gy , stronger responses to output

growth generally lower welare, while higher responses to the output gap are welare improving.

However, the quantitative changes in welare are small and neither measure plays a signifcant

role in determining the optimal ination rate within the determinacy region o the parameterspace.

While our baseline specifcation o the Taylor rule restricts the endogenous response o the

central bank to ination and the real side o the economy, an additional actor sometimes con-

sidered is PLT. While the evidence or central banks actually ollowing PLT remains scarce,PLT has nonetheless received substantial attention in the literature or several reasons. First, asemphasized in Woodord (2003), PLT guarantees determinacy under zero trend ination or any

positive response to the price-level gap. Second, Coibion and Gorodnichenko (2011) show that

PLT ensures determinacy or > 0 as well and is not subject to the deterioration o the Taylor

principle as a result o positive trend ination which occurs when the central bank responds only

to ination. Third, Gorodnichenko and Shapiro (2007) show that PLT robustly helps to stabilizeination expectations, thereby yielding smaller ination and output volatility than would occur

in ination-targeting regimes.

We extend our baseline model to include PLT in the central banks reaction unction (p > 0).

Panel F o Figure 9 shows the eects o PLT on welare or dierent as well as its impli-

cations or the optimal rate o ination. First, PLT strictly increases welare or any , espe-cially at low levels o ination. Second, PLT leads to much lower levels o optimal ination

than ination-targeting regimes. Even or moderate responses to the price-level gap, the opti-

mal level o ination is

7/28/2019 82770969.pdf

33/37


stability (rather than ination stability) which is, in act, the mandated objective or most centralbanks.

The intuition or why PLT delivers such a small optimal ination rate is straightorward.

First, as observed in Gorodnichenko and Shapiro (2007), PLT stabilizes expectations and has a

proound eect on output and ination volatility. In our simulations, the reduction in ination

and output volatility is so substantial that the welare costs o ination are almost exclusivelydriven by the steady-state eects. As a result o reduced volatility, the ZLB binds less requently.

For example, with p = 03, the ZLB binds

7/28/2019 82770969.pdf

34/37


normative point o view, we also show that welare could be substantially improved by introduc-ing PLT. The latter helps to stabilize economic uctuations and reduces the probability o hitting

the ZLB. As a result, the optimal ination rate under a PLT regime would be close to zero. In

other words, optimal monetary policy, characterized as a combination o a low ination target

and a systematic response o nominal interest rates to deviations o the price level rom its target,

can be interpreted as being very close to the price stability enshrined in the legal mandates omost central banks.

In the absence o such a change in the interest rate rule, our results suggest that higher in-

ation targets are likely too blunt an instrument to address the ZLB in a way that signifcantly

increases aggregate welare. This is not because ZLB episodes are not costly, but rather because

the perpetual costs o higher ination outweigh the benefts o more inrequent ZLB episodes.Addressing the very large costs associated with ZLB episodes is thereore likely to require alter-

native policies more explicitly ocused on these specifc episodes, such as countercyclical fscal

policies or the use o non-standard monetary policy tools. The lack o consensus on the ef-

cacy o these policy tools, however, suggests that they should be high on the research agenda o

macro-economists.

Acknowledgments. We are grateul to anonymous reerees, the Editor, Roberto Billi, Ariel Burstein, GautiEggertsson, Jordi Gali, Marc Gianonni, Christian Hellwig, David Romer, Eric Sims, Alex Wolman, and seminar partici-pants at John Hopkins University, Bank o Canada, College o William and Mary, NBER Summer Institute in MonetaryEconomics and Economic Fluctuations and Growth, University o Wisconsin, Society or Computational Economics,Richmond Fed and UNC Chapel Hill or helpul comments. Gorodnichenko thanks the National Science Foundation orfnancial support.

Supplementary Data

Supplementary data are available at Review o Economic Studies online.

REFERENCES

ALTIG, D., CHRISTIANO, L., EICHENBAUM, M. and LINDE, J. (2010), Firm-Specifc Capital, Nominal Rigiditiesand the Business Cycle, Review o Economic Dynamics, 14, 225247.

AMANO, R. and SHUKAYEV, M. (2010), Risk Premium Shocks and the Zero Bound on Nominal Interest Rates,Journal o Money, Credit and Banking, orthcoming.

ARUOBA, S. and SCHORFHEIDE, F. (2011), Sticky Prices versus Monetary Frictions: An Estimation o Policy Trade-os, AEJ Macroeconomics, 3, 6090.

ASCARI, G. and ROPELE, T. (2007), Trend Ination, Taylor Principle, and Indeterminacy, Journal o Money, Creditand Banking, 48, 15571584.

BASU, S. and FERNALD, J. (1997), Returns t

82770969.pdf

Documents