Aggregate Implications of Lumpy Adjustment

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1 Aggregate Implications of Lumpy Adjustment Eduardo Engel Cowles Lunch. March 3rd, 2010 Eduardo Engel 1

2 1. Motivation Micro adjustment is lumpy for many aggregates of interest: stock of durable good nominal prices capital stock employment Is this relevant for aggregate dynamics? Investment: Yes... in partial equilibrium: Caballero - Engel - Haltiwanger (1995), Cooper - Haltiwanger - Power (1999), Caballero - Engel (1999) No... in general equilibrium: Thomas (2002), Khan - Thomas (2003, 2008) Yes... in general equilibrium: This paper Eduardo Engel 2

3 Relevant in what sense? Better micro foundations important per se Better match of the data Better out-of-sample forecasts Matters for relevant policy questions In this paper: IRF with signicant and systematic history dependence Eduardo Engel 3

4 In a nutshell Time series model 1961:I 1989:I 2000:II Lumpy Model 1961:I 1989:I 2000:II FL Model 1961:I 1989:I 2000:II Quarters Quarters Quarters Eduardo Engel 4

5 Outline 1. Motivation 2. Basic mechanism 3. Time series evidence 4. Model 5. Confronting the evidence 6. Aggregate dynamics 7. Conclusion Eduardo Engel 5

6 2. Basic mechanism Rationalize lumpy micro behavior via non-convex (xed) adjustment costs Need to take heterogeneous rms seriously Main ingredients: cross-section of mandated investment: f (x) inaction range: L x U f (x) and L, U depend on the state of the economy: aggregate shocks distribution of rm specic shocks distribution of capital stock Eduardo Engel 6

7 Ss policy: prob of adjusting f(x,t): x section mandated investment mandated investment rate x I + L t = x f (x, t)dx x f (x, t)dx K t U IRF 0,t = F (L) + (1 F (U)) } {{ } intensive margin + L f (L) + Uf (U) }{{} extensive margin Eduardo Engel 7

8 Lumpy Investment and Time-Varying IRFs After a sequence of above avge. shocks (`boom'): f (x, t) with more mass close to upper trigger investment more responsive to a marginal shock Similarly: investment less responsive during downturns, Continuous time for a formal result Beware of linear models when predicting the impact of a stimulus To what an extent does this intuition extend to a fully specied DSGE model? Eduardo Engel 8

9 3. Time series evidence Let: x t I t /K t 1 Consider the following GARCH-type model: x t = p φ j x t j + σ t e t, j=1 σ t = h(x t 1, x t 2, ), It follows that: IRF 0,t = x t ε t = σ t = h(x t 1, x t 2, ) Consider two specications (also kernel estimators): h(x t 1, x t 2, ) = α 0 + α 1 x k t 1, h(x t 1, x t 2, ) = ( α 0 + α 1 x k t 1) 2, with x k t 1 k k j=1 x t j. Eduardo Engel Basic mechanism predicts: α 1 > 0, α 1 > 0. 9

10 Data: private, xed, non-residential investment-to-capital-ratio quarterly, 1960.I2005.IV, BLS Series: All Equip Str p: k: α : t-α 1 : one sided p-α 1 : ±log(σ 95 /σ 5 ): ±log(σ 90 /σ 10 ): no. obs. est. p: no. obs. est. k: Eduardo Engel 10

11 4. Model Incorporates lumpy investment (and therefore rm heterogeneity) into an otherwise standard stochastic growth model Producer side: interesting Household side: simple Follows closely Khan and Thomas (2008) Two dierences: sector specic productivity shocks maintenance investment: necessary to continue operation (fraction χ of depreciated capital) Eduardo Engel 11

12 Production Units No entry or exit Aggregate, sectoral and idiosyncratic productivity shocks Unit's production function: with log-ar(1) shocks θ + ν < 1 y t = z t ɛ S,tɛ I,tk θ t nν t. I.i.d. cost of adjusting capital, ξ, drawn from a U[0, ξ], measured in units of labor Eduardo Engel 12

13 Production Units: Bellman Equation Unit's problem: V 1 (ɛ S, ɛ I, k, ξ; z, µ) = max {CF + n max(v i, max[ AC + V a ])}, k where CF = [zɛ S ɛ I k θ n ν ω(z, µ)n i M ]p(z, µ), V i = βe[v 0 (ɛ S, ɛ I, ψ(1 δ)k/γ; z, µ )], AC = ξω(z, µ)p(z, µ), V a = ip(z, µ) + βe[v 0 (ɛ S, ɛ I, k ; z, µ )], µ = distribution of (ɛ S, ɛ I, k). Eduardo Engel 13

14 Households A continuum of identical households with access to a complete set of state-contingent claims Felicity function: The intertemporal price: U(C, N h ) = log C AN h p(z, µ) U C (C, N h ) = 1/C (z, µ). The intratemporal price: ω(z, µ) U N(C, N h ) p(z, µ) = A p(z, µ). Eduardo Engel 14

15 Recursive Equilibrium A recursive competitive equilibrium is a set of functions such that ω, p, V 1, N, K, C, N h, Γ 1. Production unit optimality: Taking ω, p and Γ as given, demand N and K 2. Household optimality: Taking ω and p as given, the household optimally chooses consumption C and labor N h 3. Commodity market clearing 4. Labor market clearing 5. Model consistent dynamics: µ = Γ(z, µ). Eduardo Engel 15

16 Equilibrium Computation µ: innite dimensional We follow Krusell and Smith: approximate µ by its rst moment over capital approximate µ = Γ(z, µ) by a log-linear rule To simplify computations: ρ S = ρ I, the unit then only cares about ɛ ɛ S ɛ I. Eduardo Engel 16

17 5. Confronting the evidence Most parameters: standard values suggested by micro studies No such values available for the adjustment cost parameter ξ and the maintenance parameter χ Some options: Maximum likelihood? Match certain moments Moments from the distribution of plant level investment? how many micro units in the model correspond to one observed micro unit? Moments suitable to gauge the relative importance of PE and GE smoothing Eduardo Engel 17

18 Sources of Smoothing in Macroeconomics Aggregate Shocks micro frictions price responses Macro Aggregates Eduardo Engel 18

19 Sources of Smoothing: Lumpy Investment Models 1. Micro frictions PE smoothing: it isn't only the size of adjustment costs aggregation is central Caplin and Spulber (1987) as an extreme example 2. Price responses GE smoothing: quasi labor supply supply of funds Eduardo Engel 19

20 Our Calibration There are many combinations of PE and GE smoothing that achieve the same degree of aggregate smoothing Use 3-digit sectoral data to calibrate the relative importance of PE and GE smoothing mainly partial equilibrium eects at this level: Benchmark calibration: Match: σ sect (I /K ), σ agg (I /K ), ±log(σ 95 /σ 5 ) Parameters: ξ, χ, σ A Robustness check: Match: σ sect (I /K ), σ agg (I /K ) Parameters: ξ, σ A Eduardo Engel 20

21 Standard Choices Model period: quarter Standard choices: β = , δ = 0.022, ρ A = 0.95,... ν = 0.64 and θ = 0.18: labor share: 0.64 revenue-elasticity of capital: 0.50 σ S = , ρ S = : standard Solow residual calculation on annual 3-digit manufacturing data, taking into account sector-specic trends and time-aggregation σ I = total s.d. = 0.10 Non-trivial choices: ξ and χ Eduardo Engel 21

22 Results: Economic Magnitude of Adjustment Costs Tot. adj. costs/ Tot. adj. costs/ Adj. costs/ Adj. costs/ Agg. Output Agg. Invest. Unit Output Unit Wage Bill quart. 0.35% 2.41% 9.53% 14.88% annual 0.41% 2.84% 3.60% 5.62% Eduardo Engel 22

23 Smoothing and σ(i /K): RBC No frictions only PE only GE 0% 100% PE and GE 100% Eduardo Engel 23

24 Smoothing and σ(i /K): Khan and Thomas No frictions only PE only GE 16.1% 100% PE and GE 100% Eduardo Engel 24

25 Smoothing and σ(i /K): This Paper No frictions only PE only GE 81.0% 84.6% PE and GE 100% Eduardo Engel 25

26 Why the Dierence? We choose to match sectoral investment volatility: 3-dig. Agg. Ratio Data 1.66 This paper: 1.66 Frictionless/Khan-Thomas (2008): We choose to match IRF volatility: log(σ 95 /σ 5 ) Data 0.30 This paper: 0.29 Frictionless/Khan-Thomas (2008): 0.05 Eduardo Engel 26

27 6. Aggregate Investment Dynamics Explain why our DSGE model with lumpy adjustment generates a procyclical IRF Is it variations in PE or GEsmoothing? Is it related to the intensive or extensive margin? Relation to the basic mechanism Eduardo Engel 27

28 Responsiveness Index Dene: I + (µ t, log z t ) I (µ t, log z t ) [ ] I K (µ t, log z t + σ A ) I K (µ t, log z t ) /σ A, [ ] I K (µ t, log z t σ A ) I K (µ t, log z t ) /( σ A ), Responsiveness Index at time t dened as: RI t 0.5 [ I + (µ t, log z t ) + I (µ t, log z t ) ]. Eduardo Engel 28

29 IRF upon impact from model ( ) Time series model Frictionless model Lumpy model Responsiveness index Quarter Eduardo Engel 29

30 Robustness check Second calibration Lumpy Baseline Lumpy 25% Maintenance Lumpy Zero Maintenance FL Log Deviations from Average RI Eduardo Engel 30

31 Why strongly procyclical? A decline in the strength of PE-smoothing explains the rise in the index during the boom phase the responsiveness index uctuates much less in the frictionless economy frictionless economy only has GE-smoothing hence: contribution of GE smoothing to uctuations in responsiveness index of lumpy economy is small As the boom proceeds, the economy comes closer to the Caplin-Spulber limit Eduardo Engel 31

32 Mandated Investment We have: k = k (ɛ; z, k), (1 δ + χδ)k, otherwise. if ξ ξ T (ɛ, k; z, k), We dene mandated investment for a unit with current state (ɛ, z, k) and current capital k as: x(ɛ; z, k) log k (ɛ; z, k) log[1 δ + χδ]k. Eduardo Engel 32

33 Mandated Investment Cross-Section and Hazard St. State (I/89) Boom(II/00) Bust(I/61) Mandated Investment 0 Eduardo Engel 33

34 Why strongly procyclical? During booms: the fraction of units with mandated investment close to zero decreases the fraction of units with mandated investment above 40% increases the fraction of units with negative mandated investment decreases the x-section moves intro regions where the probability of adjusting is higher and steeper this eect is not present in a frictionless (or Calvo) model Eduardo Engel 34

35 RI: Intensive and Extensive Margins RI RI due to Fraction of Lumpy Investors RI due to Average Lumpy Investment Rate Quarters Fluctuations in responsiveness index driven mainly by variations in the fraction of units adjusting (extensive margin) Eduardo Engel 35

36 I /K : Intensive and Extensive Margins Fraction of Lumpy Adjusters Average Lumpy Investment Rate Log Deviations from the Average Log Deviations from the Average Quarters Quarters Doms and Dunne (1998): it's the fraction of units undergoing major investment episodes Eduardo Engel 36

37 Understanding the Bust 0.05 Lumpy FL More capital accumulation in the lumpy economy Large fraction in region where units are unresponsive to shocks Eduardo Engel 37

38 7. Conclusion Time-series evidence suggests time-varying IRFs Lumpy adjustment DSGE models with mainly GE-smoothing forces cannot deliver history dependent IRFs Lumpy adjustment DSGE models where both PE and GE-smoothing are relevant deliver can Eduardo Engel 38

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