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Mathias Trabandt (Freie Universität Berlin) Resolving the Missing Deflation Puzzle

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Jesper Lindé Mathias Trabandt
Sveriges Riksbank Freie Universität Berlin
February 28, 2019 Eesti Pank

Veröffentlicht in: Wirtschaft & Finanzen
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Mathias Trabandt (Freie Universität Berlin) Resolving the Missing Deflation Puzzle

  1. 1. Resolving the Missing De‡ation Puzzle Jesper Lindé Mathias Trabandt Sveriges Riksbank Freie Universität Berlin February 28, 2019
  2. 2. Motivation Key observations during the Great Recession: Extraordinary contraction in GDP but only small drop in in‡ation. Source: Christiano, Eichenbaum and Trabandt (2015, AEJ: Macro)
  3. 3. Motivation Small drop in in‡ation referred to as the “missing de‡ation 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 in‡ation] 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 in‡ation to fall by twice as much as it has.”
  4. 4. Motivation Recent work emphasizes role of …nancial frictions to address the missing de‡ation 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.
  5. 5. What We Do Study in‡ation 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. Kimball (1995) state-dependent demand elasticity. Study implications for: Propagation of shocks Nonlinear Phillips curves Unconditional distribution (skewness) of in‡ation
  6. 6. Framework Benchmark model: Erceg-Henderson-Levin (2000) model. Nominal rigidities in prices and wages. ZLB constraint on nominal interest rate. Estimated model: Christiano-Eichenbaum-Evans (2005)/Smets Wouters (2007) model.
  7. 7. Outline Benchmark model Parameterization Results Analysis in estimated model Conclusions
  8. 8. Model: Households Household j preferences: max E0 ∞ ∑ t=0 βt ςt ( ln Cj,t N 1+χ j,t 1 + χ ) ςt discount factor shock. Budget constraint: Pt Cj,t + Bj,t = Wj,t Nj,t + (1 + it 1) Bj,t 1 + Θj,t
  9. 9. Model: Households Standard Euler equation 1 = βEt δt+1 1 + it 1 + πt+1 Ct Ct+1 δt+1 ςt+1 ςt where δt follows an AR(1) process. Calvo sticky wages (same conceptual setup as for sticky prices, discussed next).
  10. 10. Model: Final Good Firms Competitive …rms aggregate intermediate goods Yi,t into …nal good Yt using technology R 1 0 G (Yi,t /Yt ) di = 1. Following Dotsey-King (2005) and Levin-Lopez-Salido-Yun (2007): G Yi,t Yt = ωp 1 + ψp 1 + ψp Yi,t Yt ψp 1 ωp + 1 + ψp ωp 1 + ψp ψp < 0: Kimball (1995), ψp = 0: Dixit-Stiglitz. Kimball aggregator: demand elasticity for intermediate goods increasing function of relative price. Dampens …rms’price response to changes in marginal costs.
  11. 11. Demand Curves: Kimball vs. Dixit-Stiglitz 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Relative Demand Yi / Y, log-scale 0.98 0.985 0.99 0.995 1 1.005 1.01 1.015 1.02 RelativePricePi /P,log-scale Demand Curves Dixit-Stiglitz ( p =0) Kimball ( p =-3) Kimball ( p =-12)
  12. 12. Model: Intermediate Good Firms Continuum of monopolistically competitive …rms i : Production technology: Yi,t = A N1 α i,t Calvo sticky prices
  13. 13. Model: Aggregate Resources Aggregate resource constraint: Ct = Yt 1 pt (wt )1 α N1 α t pt and wt are price and wage dispersion terms (Yun 1996).
  14. 14. Model: Monetary Policy Taylor rule: 1 + it = max 1, (1 + i) 1 + πt 1 + π γπ Yt Y pot t γx Y pot t denotes ‡ex price-wage output. Taylor rule in “linearized” model: it i = max f i, γπ (πt π) + γx xt g
  15. 15. 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 de…nition). 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).
  16. 16. Parameterization Price setting: 3 quarter price contracts (ξp = 0.67), 10% markup (φp = 1.1). Kimball (ψp = 12.2) and Discounting (β = 0.9975) so that κp (1 ξp )(1 βξp ) ξp 1 1 φp ψp = 0.012 in ˆπt = βEt ˆπt+1 + κp cmct (Gertler-Gali 1999, Sbordone 2002, Altig-Christiano-Eichenbaum-Lindé 2011). Wage setting: ξw = 0.75, φw = 1.1 and ψw = 6 (approx. estimate in estimated model).
  17. 17. Parameterization Labor share = 0.7 (α = 0.3), linear labor disutil. (χ = 0) Steady state in‡ation 2 percent, nominal interest rate 3 percent. Taylor rule: γπ = 1.5, γx = 0.125. δt follows AR(1) with ρ = 0.95
  18. 18. Results: E¤ects of a Discount Factor Shock Follow ZLB literature: assume negative demand shock hits economy. Discount factor shock δt rises by 1 percent before gradually receding.
  19. 19. 0 5 10 15 -2 -1 0 1 2 3 AnnualizedPercent Nominal Interest Rate 0 5 10 15 -0.5 0 0.5 1 1.5 2 AnnualizedPercent Inflation 0 5 10 15 Quarters -6 -4 -2 0 Percent Output Gap 0 5 10 15 -2 -1 0 1 2 3 Nominal Interest Rate 0 5 10 15 -0.5 0 0.5 1 1.5 2 Inflation Figure 2: Impulse Responses to a 1% Discount Factor Shock 0 5 10 15 Quarters -6 -4 -2 0 Output Gap Panel A: ZLB Not Imposed Panel B: ZLB Imposed Nonlinear Model Linearized Model
  20. 20. Results: Intuition What accounts for the muted in‡ation response in the nonlinear version of the New Keynesian model? Key: nonlinearity of Kimball aggregator and …rms’demand curve. Recall: demand elasticity falls when relative price of a …rm falls. Thus, …rms’ability to increase demand by cutting price is limited. Large price cut results in lower pro…ts because demand would increase only by little -> little incentive for …rms to cut prices a lot. Nonlinearity stronger the deeper the recession. Linearization produces large approximation errors.
  21. 21. 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: δt δ = 0.95 (δt 1 δ) + σεt with εt N(0, 1) Set σ such that prob(ZLB) = 0.10 in both models.
  22. 22. 2000 4000 6000 8000 10000 0 5 10 15 AnnualizedPercent Nominal Interest Rate 2000 4000 6000 8000 10000 -20 -10 0 10 Percent Output Gap 2000 4000 6000 8000 10000 -5 0 5 AnnualizedPercent Inflation 2000 4000 6000 8000 10000 0 5 10 15 Nominal Interest Rate 2000 4000 6000 8000 10000 -20 -10 0 10 Output Gap 2000 4000 6000 8000 10000 -5 0 5 Inflation Figure 3: Stochastic Simulation of Nonlinear and Linearized Model Panel A: Nonlinear Model Panel B: Linearized Model
  23. 23. Results: Phillips Curve
  24. 24. Results: Wage Phillips Curve
  25. 25. Analysis in Estimated Model Assess robustness in CEE (2005) - SW (2007) 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
  26. 26. Analysis in Estimated Model Estimate linearized model on standard macro data (SW 2007) Output, consumption, investment, hours, price in‡ation, wage in‡ation, federal funds rate. Pre-crisis sample: 1965Q1-2007Q4. Same seven shocks as in SW (2007). Estimate 27 parameters Calibrate price and wage stickiness parameters (ξp = 0.66 and ξw = 0.75) and markups (φp=φw =1.1). Estimate Kimball parameters ψp (post. mean -12.5) and ψw (post. mean -8.3).
  27. 27. Analysis in Estimated Model: Great Recession Next, examine model’s ability to resolve ‘missing de‡ation puzzle’. Subject nonlinear and linearized model to risk premium shock: Risk premium shock as in Smets-Wouters (2007). Bondholding FOC: λt = βEt λt+1 RP,t Rt Πt+1 . RP,t elevated for 16 quarters before gradually receding. Increase RP,t such that both models deliver a fall in GDP as in the data. Compare resulting paths of model and data for in‡ation.
  28. 28. Analysis in Estimated Model: Great Recession 2009 2011 2013 2015 -10 -5 0 GDP (%) 2009 2011 2013 2015 -2 -1 0 1 Inflation (p.p., y-o-y) 2009 2011 2013 2015 -1.5 -1 -0.5 0 Federal Funds Rate (ann. p.p.) 2009 2011 2013 2015 -10 -5 0 Consumption (%) 2009 2011 2013 2015 -30 -20 -10 0 Investment (%) 2009 2011 2013 2015 -4 -2 0 Employment (p.p.) 2009 2011 2013 2015 -4 -3 -2 -1 0 Wage Inflation (y-o-y, p.p., Earnings) Figure 5: The U.S. Great Recession: Data vs. Estimated Medium-Sized Model Notes: Data and model variables expressed in deviation from no-Great Recession baseline. Data from Christiano, Eichenbaum and Trabandt (2015) Data (Min-Max Range) Data (Mean) Nonlinear Model Linearized Model
  29. 29. Analysis in Estimated Model: Great Recession Next, study implications of nonlinear and linearized model for the Phillips curve. Using estimated parameters, simulate model for each of the seven exogenous processes separately.
  30. 30. Analysis in Estimated Model: Phillips Curves
  31. 31. Analysis in Estimated Model: In‡ation Densities Core PCE Inflation -5 0 5 10 15 Annualized Percent 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Density Wage Inflation (Hourly Earnings) -5 0 5 10 15 Annualized Percent 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Figure 8: Densities of Data vs. Stochastic Model Simulations Data (1965Q1-2007Q4) Linearized Medium-Sized Model Nonlinear Medium-Sized Model
  32. 32. 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 de‡ation puzzle’while the linearized version fails to do so. Our nonlinear model generates nonlinear Phillips curves and reproduces the skewness of price and wage in‡ation 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.
  33. 33. Additional Slides
  34. 34. Details on Kimball Aggregator Dotsey and King (2005) and Levin, Lopez-Salido and Yun (2007): G (Yi,t /Yt ) = ωp 1 + ψp h 1 + ψp (Yi,t /Yt ) ψp i 1 ωp + 1 + ψp ωp 1 + ψp G ( ) strictly concave and increasing function. ωp = 1+ψp 1+φp ψp φp, φp > 1 gross price markup, ψp 0 Kimball param. Special case: ψp = 0 ! Dixit-Stiglitz.
  35. 35. Kimball FOC’s: First order conditions can be written as: Demand Curve : Yi,t /Yt = 1 1+ψp (Pi,t /(Pt ϑp t )) εp + ψp 1+ψp Agg. Price Index: 1 = Z 1 0 (Pi,t /(Pt ϑp t )) εp di Zero Pro…ts : ϑp t = 1 + ψp ψp Z 1 0 Pi,t /Pt di εp = φp (1+ψp ) 1 φp . ϑp t –Lagr.-multiplier on aggregator constraint. Special case: ψp = 0 ! standard Dixit-Stiglitz expressions: Yi,t /Yt = (Pi,t /P) φp φp 1 , Pt = Z P 1 1 φp i,t di 1 φp
  36. 36. Intuition: Kimball vs. Dixit-Stiglitz Period-by-period pro…t function of intermediate good …rm i: Φi (pi ) = (pi mc) yi (pi ) subject to demand yi (pi ) = 1 1 + ψp p (1+θp )(1+ψp ) θp i + ψp ! y where pi denotes the relative price of …rm i.
  37. 37. Intuition Kimball vs. Dixit-Stiglitz 0.98 0.99 1 1.01 1.02 Rel. Price p i = P i / P, log-scale 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2Rel.DemandYi /Y,log-scale Demand Curves Dixit-Stiglitz ( p =0) Kimball ( p =-12) 0.99 0.995 1 1.005 1.01 Relative Price p i 0.084 0.086 0.088 0.09 0.092 Profits Profit Functions Dixit-Stiglitz ( p = 0) Kimball ( p =-12) 0.95 1 1.05 Relative Price p i 0 0.05 0.1 0.15Profits Dixit-Stiglitz: Profit Functions pi opt =1.000 pi opt =0.950 pi opt =1.050Baseline Marginal Cost -5% Marginal Cost +5% 0.995 1 1.005 Relative Price p i 0.05 0.1 0.15 Profits Kimball: Profit Functions pi opt =1.0000 pi opt =0.9970 pi opt =1.0045 Baseline Marginal Cost -5% Marginal Cost +5% Figure 1: Kimball vs. Dixit-Stiglitz -- Demand Curves, Profits and Optimal Prices
  38. 38. Intuition: Kimball vs. Dixit-Stiglitz Dixit-Stiglitz (ψp = 0), optimal price: Nonlinear : pi = (1 + θp ) mc Linearized : pi ¯pi ¯pi = mc mc mc Kimball (ψp < 0), optimal price: Nonlinear : pi 1 γp (1+θp )(1+ψp ) θp i ! = 1 + ψp (1 + θp ) ψp + θp ψp + 1 mc Linearized : pi ¯pi ¯pi = 1 1 (1 + θp ) ψp mc mc mc where γ = θp ψp ψp +θp ψp +1 .
  39. 39. Intuition Kimball vs. Dixit-Stiglitz 0.95 1 1.05 Relative Price p i 0 0.05 pi =1.050Baseline Marginal Cost +5% 0.995 1 1.005 Relative Price p i 0.05 pi opt =1.0045 Marginal Cost +5% +10 +5 0 -5 -10 -15 % Change in Marginal Cost -1 -0.5 0 0.5 1 %Changeinpi opt Kimball: Optimal Relative Price Nonlinear Model Linearized Model +10 +5 0 -5 -10 -15 % Change in Marginal Cost -10 -5 0 5 10 %Changeinpi opt Dixit-Stiglitz: Optimal Relative Price Nonlinear Model Linearized Model
  40. 40. Intuition Kimball vs. Dixit-Stiglitz Kimball (ψp < 0): asymmetric desire to adjust prices change of demand smaller for price cuts than for price hikes (in absolute value); see also next slide. …rm pro…ts: Φi (pi ) = (pi mc) yi (pi ) in response to changes in marginal costs, …rms desire price cuts to be smaller than price hikes (in absolute value). nonlinear and linearized model di¤erent countercyclical markup Dixit-Stiglitz (ψp = 0) : symmetric desire to adjust prices due to log-linear demand nonlinear and linearized model identical constant markup
  41. 41. Demand Curves: Kimball vs. Dixit-Stiglitz 0.985 0.99 0.995 1 1.005 1.01 1.015 1.02 Relative Price P i / P, log-scale 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 RelativeDemandYi /Y,log-scale Demand Curves Dixit-Stiglitz ( p =0) Kimball ( p =-12) Kimball ( p =-50)
  42. 42. Linearized Price and Wage Phillips Curves Price In‡ation Phillips curve: ˆΠt = βEt ˆΠt+1 + 1 ξp 1 βξp ξp 1 1 φp ψp cmct φp > 1 (gross price markup), ψp 0 (Kimball parameter prices). Wage In‡ation Phillips curve: ˆΠw t = βEt ˆΠw t+1 + (1 ξw ) (1 βξw ) ξw 1 1 φw ψw + χ φw φw 1 (dmrst ˆwt ) φw > 1 (gross wage markup), ψw 0 (Kimball parameter wages), χ 0 (inv. Frisch elast.).
  43. 43. Discount Factor Shock: Kimball vs. Dixit-Stiglitz Set ψp = ψw = 0 ! Dixit-Stiglitz. Set ξp = ξw = 0.9 to keep slopes of linearized Phillips curves unchanged. Kimball Dixit-Stiglitz Kimball aggregator crucial for muted in‡ation response.
  44. 44. Benchmark Model: Large vs. Small Shocks Benchmark Shock Size Small Shock Size
  45. 45. 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, νt : (νt ν) = 0.80 (νt 1 ν) + σνεν,t , εν,t iid N(0, 1) Parameter σν tuned such that probability of being the ZLB is roughly 10 percent in each quarter (Nakata, 2016).
  46. 46. Stochastic Model Solution (Lindé-Trabandt, 2018)
  47. 47. Stochastic Model Solution (Lindé-Trabandt, 2018) Results indicate that implications of uncertainty in nonlinear model are quantitatively small/negligible. Reason: ‡at 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.
  48. 48. Analysis in Est. Model: Phillips Curves Cond. on 8Q ZLB

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