Resolving the Missing Deflation Puzzle Jesper Lindé

Mathias Trabandt

Sveriges Riksbank

Freie Universität Berlin

June 26, 2018

Motivation Key observations during the Great Recession: Extraordinary contraction in GDP but only small drop in inflation.

Source: Christiano, Eichenbaum and Trabandt (2015, AEJ: Macro)

Motivation

Small drop in inflation referred to as the “missing deflation puzzle”: Hall (2011), Ball and Mazumder (2011), Coibion and Gorodnichenko (2015), King and Watson (2012), Fratto and Uhlig (2018).

John C. Williams (2010, p. 8): “The surprise [about inflation] is that it’s fallen so little, given the depth and duration of the recent downturn. Based on the experience of past severe recessions, I would have expected inflation to fall by twice as much as it has.”

Motivation

Recent work emphasizes role of financial frictions to address the missing deflation puzzle: Del Negro, Giannoni and Schorfheide (2015), Christiano, Eichenbaum and Trabandt (2015), Gilchrist, Schoenle, Sim and Zakrajsek (2017).

We propose an alternative resolution of the puzzle: Importance of nonlinearities in price and wage-setting when the economy is exposed to large shocks.

What We Do

Study inflation and output dynamics in linearized and nonlinear formulations of the standard New Keynesian model. Key feature: real rigidity to reconcile macroevidence of low Phillips curve slope and microevidence of frequent price re-setting. Real rigidity: Kimball (1995) state-dependent demand elasticity.

Study implications for: Propagation of shocks Nonlinear Phillips curves Unconditional distribution of inflation (skewness)

Framework

Benchmark model: Erceg-Henderson-Levin (2000) model. Monopolistic competition and Calvo sticky prices and wages. Fixed aggregate capital stock. ZLB constraint on nominal interest rate.

Estimated model: Christiano-Eichenbaum-Evans (2005)/Smets and Wouters (2007) model with endogenous capital.

Outline

Benchmark model Parameterization Results Analysis in estimated model Conclusions

Model: Households

Household j preferences: •

max E0

 bt Vt

t =0

(

ln Cj ,t − w

1 +c )

Nj ,t

1+c

Vt − discount factor shock. Budget constraint: Pt Cj ,t + Bj ,t = Wj ,t Nj ,t + RtK Kj + (1 + it −1 ) Bj ,t −1 + Gj ,t + Aj ,t

Model: Households

Standard Euler equation 1 = bEt d t +1 ≡

V t +1 Vt

#

1 + it Ct d t +1 1 + p t +1 Ct +1

$

where dt follows an AR(1) process.

Calvo sticky wages (same conceptual setup as for sticky prices, discussed next).

Model: Final Good Firms Competitive firms aggregate intermediate goods Yf ,t into final good R1 Yt using technology 0 G (Yf ,t /Yt ) df = 1. Following Dotsey-King (2005) and Levin-Lopez-Salido-Yun (2007): G

#

Yf ,t Yt

$

&' ) w1 ( #Y $ 1 + yp − w p p wp f ,t = 1 + yp − yp + 1 + yp Yt 1 + yp

yp < 0: Kimball (1995), yp = 0: Dixit-Stiglitz. Kimball aggregator: demand elasticity for intermediate goods increasing function of relative price. Dampens firms’ price response to changes in marginal costs.

Levin, Lopez-Salido and Yun (2007) Demand Curves

1.02

Relative Price Pf / P, log-scale

1.015

Dixit-Stiglitz (

p

=0)

1.01 1.005

Kimball (

p

=-12)

1 0.995 0.99 0.985 Kimball ( 0.98

0.85

0.9

0.95

1

1.05

1.1

Relative Demand yf / y, log-scale

p

1.15

=-3) 1.2

Model: Intermediate Good Firms

Continuum of monopolistically competitive firms f Hire workers and rent capital; production technology Yf ,t = Kfa Nf1,t−a Calvo sticky prices: optimal price set with probability 1 − x p , otherwise simple updating P˜ t = (1 + p ) Pt −1 .

Fixed aggregate capital stock K ≡

R

Kf df .

Model: Aggregate Resources

Aggregate resource constraint: Ct = Yt ≤

1 pt∗

(wt∗ )1 −a

K a Nt 1 − a

where pt∗ and wt∗ are Yun’s (1996) aggregate price and wage dispersion terms.

Model: Monetary Policy

Taylor rule: *

1 + it = max 1, (1 + i )

#

1 + pt 1+p

$ gp #

Yt Ytpot

where Ytpot denotes flex price-wage output. Taylor rule in “linearized” model: it − i = max {−i, gp (p t − p ) + gx xt }

$ gx +

Solving the Model

Solve linearized and nonlinear model using Fair-Taylor (1983, ECMA): Two-point boundary value problem. Solution of nonlinear model imposes certainty equivalence (just as linearized model solution does by definition). Solution algorithm traces out implications of not linearizing equilibrium equations. Use Dynare for computations: ‘perfect foresight solution’/‘deterministic simulation’.

Robustness: global stochastic solution, see Lindé-Trabandt (2018).

Parameterization

Price setting: x p = 0.67 (3 quarter price contracts), fp = 1.1 (10% markup). yp = −12.2 (Kimball) and b = 0.9975 (discounting) so that (1 −x p )(1 − bx p ) 1 ˆ t = bEt pˆ t +1 + k p mc kp ≡ ct 1 −f y = 0.012 in p x p

p

p

(Gertler-Gali 1999, Sbordone, 2002, ACEL 2011).

Wage setting: x w = 0.75, fw = 1.1 and yw = −6 (approx. estimate in estimated model).

Parameterization

Labor share = 0.7 (a = 0.3), linear labor disutil. (c = 0) Steady state inflation 2 percent, nominal interest rate 3 percent. Taylor rule: gp = 1.5, gx = 0.125. dt follows AR(1) with r = 0.95

Results: E§ects of a Discount Factor Shock

Follow ZLB literature: assume negative demand shock hits economy. Discount factor shock dt rises by 1 percent before gradually receding.

Figure 2: Impulse Responses to a 1% Discount Factor Shock Panel A: ZLB Not Imposed

Panel B: ZLB Imposed

Nonlinear Model

Nominal Interest Rate

Annualized Percent

3 2

2

1

1

0

0

-1

-1

-2 5

10

15

Inflation

2

0

10

15

Inflation

1.5

1

1

0.5

0.5

0

0 0

5

10

15

Output Gap

0

Percent

5

2

1.5

-0.5

Nominal Interest Rate

-2 0

Annualized Percent

Linearized Model 3

-0.5

0

5

-2

-4

-4

-6

15

Output Gap

0

-2

10

-6 0

5

10 Quarters

15

0

5

10 Quarters

15

Results: Intuition

What accounts for the muted inflation response in the nonlinear version of the New Keynesian model? Key: nonlinearity of Kimball aggregator and firms’ demand curve. Recall: demand elasticity falls when relative price of a firm falls. Thus, firms’ ability increase demand by cutting price is limited. Large price cut results in lower profits because demand would increase only by little -> little incentive for firms to cut prices a lot.

Nonlinearity stronger the deeper the recession. Linearization produces large approximation errors.

Results: Stochastic Simulations

Next, do stochastic simulations of linearized and nonlinear model using discount factor shocks. Subject both models to long sequence of discount factor shocks: dt − d = 0.95 (dt −1 − d) + s#t with #t ∼ N (0, 1) Set s such that prob(ZLB) = 0.10 in both models.

Annualized Percent

Figure 3: Stochastic Simulation of Nonlinear and Linearized Model Panel A: Nonlinear Model

Panel B: Linearized Model

Inflation

Inflation

5

5

0

0

-5

-5 2000

6000

8000

10000

Output Gap

10

Percent

4000

2000

10

0

0

-10

-10

-20

-20 2000

4000

6000

8000

10000

2000

Annualized Percent

Nominal Interest Rate

4000

6000

8000

10000

8000

10000

Output Gap

4000

6000

Nominal Interest Rate

15

15

10

10

5

5

0

0 2000

4000

6000

8000

10000

2000

4000

6000

8000

10000

Results: Phillips Curve

Results: Wage Phillips Curve

Analysis in Estimated Model

Assess robustness in CEE/SW workhorse model. Key model features: Nominal price and wage stickiness Kimball aggregation in prices and wages Endogenous capital accumulation Habit persistence and investment adjustment costs Variable capital utilization

Analysis in Estimated Model

Estimate linearized model on standard macro data (SW 2007) Output, consumption, investment, hours worked per capita, inflation, wage inflation and federal funds rate. Pre-crisis sample: 1965Q1-2007Q4. Same seven shocks as in SW (2007).

Estimate 27 parameters Calibrate price and wage stickiness parameters (x p = 0.66 and x w = 0.75) and markups (fp =fw =1.1). Estimate Kimball parameters yp (post. mean -12.5) and yw (post. mean -8.3).

Analysis in Estimated Model: Great Recession

Next, we aim to examine the model’s ability to shed light on the ‘missing deflation puzzle’. Subject nonlinear and linearized model to risk premium shock: Risk premium shock as in Smets-Wouters (2007). Bondholding FOC: e ,t R t lt = bEt lt +1 RP P t +1 . eRP ,t elevated for 16 quarters before gradually receding. Increase eRP ,t such that both models deliver a fall in output as in the data. Compare resulting paths of model and data for inflation.

Analysis in Estimated Model: Great Recession Figure 5: The U.S. Great Recession: Data vs. Estimated Medium-Sized Model Data (Min-Max Range)

Data (Mean)

GDP (%)

0

Nonlinear Model

Inflation (p.p., y-o-y)

0

1

Federal Funds Rate (ann. p.p.)

-0.5

0

-5

Linearized Model

-1 -1 -10

-1.5 -2 2009

2011

2013

2015

Consumption (%)

0

2009

2011

2013

2015

2009

Investment (%)

0

2011

2013

2015

Employment (p.p.) 0

-10

-2

-5 -20

-4 -30

-10 2009

2011

2013

2015

2009

2011

2013

2015

2009

2011

2013

Wage Inflation (y-o-y, p.p., Earnings) 0 -1 -2 -3 Notes: Data and model variables expressed in deviation from no-Great Recession baseline. Data from Christiano, Eichenbaum and Trabandt (2015)

-4 2009

2011

2013

2015

2015

Analysis in Estimated Model: Great Recession

Next, study the implications of the nonlinear and linearized model for the Phillips curve. Simulate the model for each of the seven exogenous processes using the estimated model parameters.

Analysis in Estimated Model: Phillips Curves

Analysis in Estimated Model: Inflation Densities Figure 8: Densities of Data vs. Stochastic Model Simulations Data (1965Q1-2007Q4)

Core PCE Inflation

0.35

Density

Linearized Medium-Sized Model

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

Nonlinear Medium-Sized Model

Wage Inflation (Hourly Earnings)

0 -5

0

5

10

Annualized Percent

15

-5

0

5

10

Annualized Percent

15

Conclusions

Our analysis focuses on nonlinearities in price and wage-setting using Kimball (1995) aggregation. Our nonlinear NK model with Kimball aggregation resolves the ‘missing deflation puzzle’ while the linearized version fails to do so. Our nonlinear model generates nonlinear Phillips curves and reproduces the skewness of price and wage inflation observed in post-war U.S. data. All told, our results caution against the common practice of using linearized models when the economy is exposed to large shocks.

Additional Slides

Details on Kimball Aggregator

Dotsey and King (2005) and Levin, Lopez-Salido and Yun (2007): G

#

Yf ,t Yt

$

&' ) w1 ( #Y $ 1 + yp − w p p wp f ,t = 1 + yp − yp + 1 + yp Yt 1 + yp

G (·) strictly concave and increasing function. wp =

1 + yp 1 + fp yp f p ,

fp > 1 gross price markup, yp ≤ 0 Kimball param.

Special case: yp = 0 ! Dixit-Stiglitz.

Kimball FOC’s: First order conditions can be written as: :

Y f ,t Yt

Agg. Price Index :

1=

Demand Curve

=

1 1 + yp

Z 1# Pf ,t 0

Zero Profits #p =

fp (1 + yp ) 1 − fp .

:

'

( P f ,t #p P t Jpt $ #p

Pt Jpt

Jpt = 1 + yp − yp

+

yp 1 + yp

df

Z 1 0

P f ,t P t df

Jpt — Lagr.-multiplier on aggregator constraint.

Special case: yp = 0 ! standard Dixit-Stiglitz expressions: Yf ,t = Yt

#

Pf ,t Pt

$− f fp−1 p

, Pt =

#Z

1 1 − fp

Pf ,t df

$ 1 − fp

Linearized Price and Wage Phillips Curves

Price Inflation Phillips curve: . /. / 1 − x p 1 − bx p 1 ˆ t = bEt P ˆ t +1 + P mc ct xp 1 − fp yp

fp > 1 (gross price markup), yp ≤ 0 (Kimball parameter prices).

Wage Inflation Phillips curve: 1 ˆ wt = bEt P ˆ wt+1 + (1 − x w ) (1 − bx w ) P d rs t − wˆ t ) (m xw 1 − fw yw

fw > 1 (gross wage markup), yw ≤ 0 (Kimball parameter wages).

Discount Factor Shock: Kimball vs. Dixit-Stiglitz Set yp = yw = 0 ! Dixit-Stiglitz. Set x p = x w = 0.9 to keep slopes of linearized Phillips curves unchanged. Kimball

Dixit-Stiglitz

Kimball aggregator crucial for muted inflation response.

Benchmark Model: Large vs. Small Shocks Benchmark Shock Size

Small Shock Size

Stochastic Model Solution (Lindé-Trabandt, 2018)

Use global projection method to solve stochastic nonlinear model: Time iteration method (Judd, 1988, Coleman, 1990, 1991). No certainty equivalence. Discretize state space. Linear interpolation/extrapolation for points not exactly on grid nodes. Evaluate expectation terms using trapezoid rule.

Stochastic processes for consumption demand shock, nt : iid

(nt − n) = 0.80 (nt −1 − n) + sn #n,t , #n,t ∼ N (0, 1) Parameter sn tuned such that probability of being the ZLB is roughly 10 percent in each quarter (Nakata, 2016).

Stochastic Model Solution (Lindé-Trabandt, 2018)

Stochastic Model Solution (Lindé-Trabandt, 2018)

Results indicate that implications of uncertainty in nonlinear model are quantitatively small/negligible. Reason: flat Phillips curve and Kimball aggregator. Hence, we focus on the deterministic nonlinear solution.

By contrast, shock uncertainty a§ects linearized model solution to a much greater degree.

Analysis in Est. Model: Phillips Curves Cond. on 8Q ZLB