flexiparadox

State-Dependent Pricingand the Paradox of Flexibility

Luca Dedola and Anton Nakov

ECB and CEPR

September 2015

Policy rates in rich economies have been constant for years

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Constant policy rates and shock amplification

I Recent literature shows ZLB leads to amplification of shocks:

Woodford (2011), Christiano, Eichenbaum, Rebelo (2011), Del Negro et

al. (2013), Erceg and Linde (2014), Kiley (2014)

I A drop in the natural real rate has large deflationary effects

I The government spending multiplier is (much) larger than 1

I Forward guidance is extremely powerful

I “Paradoxes” of thrift, toil, and flexibility (Eggertsson-Krugman)

Constant policy rates and shock amplification

I Mechanism:

I Suppose it = i for t = 0...T and then iT+k = i + φπT+k

I Then c0 = σ∑T

t=1 E0πt + cT ≈ σ (EtpT − p0)

I Let g0 > 0

I Since policy is such that pT > p0 then c0 > 0

I Then ∂Y0

∂G0> 1

The paradox of flexibility

I “Paradox of flexibility”: with it = i increasing price flexibilityleads to a larger response of inflation and larger amplification

I I.e. it leads to a deeper recession and deflation when hit by adeflationary shock (Eggertsson and Krugman)

I Also leads to a larger government spending multiplier(Christiano et. al., Erceg and Linde)

What role for the form of nominal price rigidities?

I These papers all assume Calvo price setting

I Ohanian (2010): results depend on rigidity in Calvo model, butincentives to change prices rise during turbulent/crisis periods

I In general the details of price setting at the micro level maymatter a great deal for the dynamics of aggregate variables

I E.g. in Golosov-Lucas (2007) the price level is a lot moreflexible than Calvo (for the same adjustment frequency)

This paper

I Looks at the effects of a shock to government spending whenthe nominal interest rate is held constant for T periods

I Across three pricing models: Calvo, fixed menu cost, andencompassing model

I Encompassing model is “smoothly state-dependent” :adjustment probability is a smoothly increasing function

I Decision rules take non-linearity with respect to the microstate into account (Reiter’s method)

I We look at different monetary policy rules with/withoutconstant interest rate (CIR)

Main findings

I With constant interest rates SDP can produce even largeramplification than Calvo

I The surprising results at the ZLB are a feature of stickyprices, not just an artifact of Calvo

I The Phillips curve becomes steeper at the ZLB under SDP(and is unchanged under Calvo)

A very large literature

I ZLB/constant rate in (large) DSGE models

I Normative analyses

I Empirical studies about amplification at ZLB

I We do not pretend to say anything about what happened inreality; we only try to shed light on a theoretical mechanism

Outline of the talk

1. Introduction X

2. Model

3. Results

4. Conclusions

Modeling idiosyncratic shocks and state-dependent pricing

I Model deviates in two ways from textbook NK model:

I Idiosyncratic productivity shocks to firms

I State-dependent pricing nesting Calvo and fixed menu costs

I We focus on dynamics under a constant interest rate for Tperiods (anticipated shocks to Taylor rule, Galı 2012)

Model: households

I The household’s period utility is

C 1−γt

1− γ− χN1+ψ

t

1 + ψ+ log(Mt/Pt)

I Consumption is a CES aggregate of differentiated products

Ct =

∫ 1

0C

ε−1ε

it di

εε−1

I The household’s nominal period budget constraint is∫ 1

0PitCitdi + Mt + R−1

t Bt = WtNt + Mt−1 + Tt + Bt−1

Model: household optimality conditions

I Households choose Cit ,Nt ,Bt ,Mt to maximize expectedutility, subject to the budget constraint

I Optimal consumption across the differentiated goods

Cit = (Pt/Pit)εCt

Pt ≡[∫ 1

0Pit

1−εdi

] 11−ε

I Optimal labor supply, consumption, and money use

χCγt Nψ

t = Wt/Pt

1 = βRtEt

[PtC

−γt+1/

(Pt+1C−γt

)]Mt/Pt = Cγ

t Rt/ (Rt − 1)

Model: monopolistic competitor firms

I Firm i produces output Yit = AitNit

I Productivity is idiosyncratic, log Ait = ρA log Ait−1 + εait ,εait ∼ N(0, σ2

a)

I Firm i faces demand from households, and the government,Yit = Cit + Git

I The government’s consumption basket is also a CES,

Gt =

∫ 10 G

ε−1ε

it di

εε−1

Model: monopolistic competitor firms

I Demand curve, Yit = (Ct + Gt)Pεt P−εit

I Period profits, Uit = PitYit −WtNit

I Discount rate, Qt,t+1 = βPtC

−γt

Pt+1C−γt+1

Model: firm value function

I Value functionV (P,A,Ω) =

U(P,A,Ω) + βE

Qt,t′ [V (P,A′,Ω′) + EG (P,A′,Ω′)]∣∣A,Ω

where EG (·) is the expected gain from adjustment

EG (P,A′,Ω′) ≡ λ

[D(P,A′,Ω′)

W (Ω′)

]D(P,A′,Ω′)

D(P,A′,Ω′) ≡ maxP

V (P,A′,Ω′)− V (P,A′,Ω′)

Model: adjustment function

I λ(L) increases with the gain from adjustment L

I In particular, we postulate

λ (L) ≡ λ

(1− λ) + λ (α/L)ξ

I where L is the relevant state

I With ξ → 0, λ (L) = λ Calvo

I With ξ →∞, λ (L) = 1 L ≥ α Fixed menu cost

Model: adjustment function and histogram fit

−0.5 0 0.50

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0.25Price changes: models vs data

Den

sity

Size of log price changes

AC NielsenFMCCalvoSSDP

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1α=0.0311, λ=0.1089

Loss from inactionP

roba

bilit

y of

adj

ustm

ent

ξ=50

ξ=0.05

ξ=1

ξ=0.23 (SSDP)

Model: monetary policy and government spending

I The monetary authority follows a Taylor rule

it = i∗ + φππt +T∑j=1

εit−j

where εit−i are chosen so that it = i∗, t = 1...T

I Government spending, gt = Gt−GY

gt = ρggt−1 + εgt

with εgt ∼ N(0, σ2g )

Calibration

Discount factor β−12 = 1.04 Golosov-Lucas (2007)CRRA γ = 2 Ibid.Elast. of subst. ε = 7 Ibid.Labor supply elast. ψ = 1Inflation target Π∗ = 1 AC NielsenInflation reaction φπ = 2Length of CIR period T = 24, 36 Erceg and Linde (2014)Persistence of Gt ρG = 0.9 Ibid.Persistence of Ait ρA = 0.9 Costain-Nakov (2011)Std. dev. of Ait σA = 0.1 Ibid.State dependence ξ = 0, 0.23, 1 Ibid.Fixed menu cost α = 0.04 Ibid.Calvo frequency λ = 0.1 Nakamura-Steinsson (2008)

Preliminaries: textbook Calvo model

I The flexible price multiplier is (Woodford, 2011)

Γ =γ

γ + ψ≤ 1

I Log-linearized consumption Euler equation(σ = γ−1

)yt − gt = Et (yt+1 − gt+1)− σ (it − Etπt+1 − r)

I Phillips curve

πt = κ

∞∑j=0

βjEt(yt − Γgt),

where κ = (1− α)(1− αβ)(γ + ψ)/α

Preliminaries: textbook Calvo model with Taylor rule

I Solution under a Taylor rule

µTR = 1−

(φπ − ρ1− βρ

κ+ φC

)(1− Γ)

(1− ρ) γ +

(φπ − ρ1− βρ

κ+ φC

) < 1

I More price flexibility (higher κ) leads to smaller multiplier

The multiplier is lower under SDP with a Taylor rule

10 20 30 400

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0.2Nominal interest rate (pp, annualized)

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0.1Real interest rate (pp, annualized)

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0.103Fraction of adjusters

Months10 20 30 40

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Months10 20 30 40

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1Fiscal multiplier

Months

CalvoFMCSSDP

Calvo with constant interest rate: Woodford’s case

I General solution

yt − gt =κ (1− Γ) ρ

(1− ρ) (1− βρ) γ − κρgt + a1λ

t1 + a2λ

t2

I When denominator ∆ > 0 then λ1, λ2 > 1 and so settinga1, a2 = 0 ensures a unique bounded solution:

yt = gt +κ (1− Γ) ρ

∆gt = µZLBgt

I Paradox of flexibility: as κ ↑, then ∆→ 0+ and µZLB → +∞

I Ohanian: paradox is limited to ∆ > 0, otherwise µZLB is notwell defined due to multiplicity of equilibria

Werning’s perfect foresight solution

I Difference equations valid only up to T ; thereafter CB followsits policy rule which determines equilibrium upon liftoff:

yt = µZLBgt −κ (1− Γ) ρ

∆

1− ρβλ1

1− βλ21

(ρ

λ1

)T−tgt +

πT+1 + (1− βλ1) γ (yT+1 − gT+1)(1− βλ2

1

)γλT−t1

,

I When T sufficiently large and κ such that ∆→ 0+, thenρ < λ1 < 1 and solution close to µZLB

I For κ larger, such that ∆ < 0, ρ/λ1 > 1, backward explosion

I The multiplier grows with T , and the Paradox of Flexibilityholds for any κ

With ZLB there is even larger amplification under SDP

10 20 30 400

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10 20 30 400

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1GDP (%)

10 20 30 400

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20

30Price level (%)

10 20 30 400

0.005

0.01

0.015Nominal interest rate (pp, annualized)

10 20 30 40−4

−2

0

2Real interest rate (pp, annualized)

10 20 30 400.095

0.1

0.105

0.11Fraction of adjusters

Months10 20 30 40

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2Selection effect

Months10 20 30 40

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10Fiscal multiplier

Months

CalvoSSDP

Largest amplification under FMC

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10 20 30 400

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10 20 30 40150

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−3Nominal interest rate (pp, annualized)

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Months10 20 30 40

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Months10 20 30 40

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60Fiscal multiplier

Months

FMC

With T = 36 amplification is larger under SSDP

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10 20 30 40−2

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10 20 30 400

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4x 10

−3Nominal interest rate (pp, annualized)

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Months10 20 30 40

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Months10 20 30 40

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Months

CalvoFMCSSDP

Conclusions

I With active monetary policy under SDP the fiscal multiplier isclose to flexible-prices (closer than Calvo)

I But with a constant interest rate there is an even largeramplification of shocks under SDP than Calvo

I The largest amplification is in the Golosov-Lucas (2007) model

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flexiparadox