Durable Goods, Inventories, and the Great Moderation

Transcription

1 Durable Goods, Inventories, and the Great Moderation James A. Kahn October 2007

2 Motivation GDP volatility dropped sharply in the early 1980s, as did the volatility of many aggregates (the Great Moderation )

3 Motivation GDP volatility dropped sharply in the early 1980s, as did the volatility of many aggregates (the Great Moderation ) Considerable debate over the cause:

4 Motivation GDP volatility dropped sharply in the early 1980s, as did the volatility of many aggregates (the Great Moderation ) Considerable debate over the cause: Smaller shocks (Stock-Watson, others): Not really an explanation, hard to reconcile with non-declining nancial volatility

5 Motivation GDP volatility dropped sharply in the early 1980s, as did the volatility of many aggregates (the Great Moderation ) Considerable debate over the cause: Smaller shocks (Stock-Watson, others): Not really an explanation, hard to reconcile with non-declining nancial volatility Good policy (Gali-Gertler, others): May be contributing factor, but hasn t been shown to explain variety of facts

6 Motivation GDP volatility dropped sharply in the early 1980s, as did the volatility of many aggregates (the Great Moderation ) Considerable debate over the cause: Smaller shocks (Stock-Watson, others): Not really an explanation, hard to reconcile with non-declining nancial volatility Good policy (Gali-Gertler, others): May be contributing factor, but hasn t been shown to explain variety of facts Financial innovation (Dynan et al): No direct micro evidence?

7 Motivation GDP volatility dropped sharply in the early 1980s, as did the volatility of many aggregates (the Great Moderation ) Considerable debate over the cause: Smaller shocks (Stock-Watson, others): Not really an explanation, hard to reconcile with non-declining nancial volatility Good policy (Gali-Gertler, others): May be contributing factor, but hasn t been shown to explain variety of facts Financial innovation (Dynan et al): No direct micro evidence? Technology/Inventory Control (Kahn et al): Original model had some counterfactual implications. Evidence mainly anecdotal, indirect.

8 Motivation From Campbell, Lettau et al (2001): No postwar reduction in nancial volatility

9 Motivation From NY Times, 3/1/84

10 Motivation 20 GDP Growth, Quarterly, Annual Rate. Source: NIPA

11 Motivation Early literature on micro volatility suggested that it had been increasing over time

12 Motivation Early literature on micro volatility suggested that it had been increasing over time Comin, Philippon

13 Motivation Early literature on micro volatility suggested that it had been increasing over time Comin, Philippon Followup literature suggested otherwise, that initial ndings were due to sample selection

14 Motivation Early literature on micro volatility suggested that it had been increasing over time Comin, Philippon Followup literature suggested otherwise, that initial ndings were due to sample selection Davis, Haltiwanger et al (employment)

15 Motivation Early literature on micro volatility suggested that it had been increasing over time Comin, Philippon Followup literature suggested otherwise, that initial ndings were due to sample selection Davis, Haltiwanger et al (employment) Fama, French ( nancial)

16 Comparison of Share Weighted Sectoral Growth Rates GDP Durables Quarterly, Annual Rate. Source: NIPA Quarterly, Annual Rate. Source: NIPA Nondurables Services Quarterly, Annual Rate. Source: NIPA Quarterly, Annual Rate. Source: NIPA

17 Motivation The decline in volatility was not uniform or simultaneous across sectors or expenditure categories.

18 Motivation The decline in volatility was not uniform or simultaneous across sectors or expenditure categories.

19 Motivation The decline in volatility was not uniform or simultaneous across sectors or expenditure categories. Standard deviations of growth rates GDP Goods Durables Nondur. Serv. Y S Y S Y S Y S Y,S

20 Motivation The decline in volatility was not uniform or simultaneous across sectors or expenditure categories. Standard deviations of growth rates GDP Goods Durables Nondur. Serv. Y S Y S Y S Y S Y,S

21 28 Durable Goods Sector Volatility (5 Year Centered Rolling Standard Deviations) Annualized Quarterly Growth Rates Final Sales Volatility Production Volatility

22 Durables Volatility and Lead Time for Orders Days Pre 84 average = Post 83 average = Production Materials: Avg. Lead Time for Orders* Output Volatility (Durables)** * Source: ISM survey **5 year centered rolling variance

23 Motivation Durable Goods Inventory Sales Ratio* Real/Real Nominal/Nominal *The series is discontinuous (and overlapping for one year) due the change from SIC to NAICS industry definitions

24 An Inventory Model for Durable Goods Production to Order Materials orders Works in process O U Z D Y M M X Y F Deliveries Final Output Note: For simplicity, no materials stocks.

25 Details Continuum of symmetric rms

26 Details Continuum of symmetric rms Technology

27 Details Continuum of symmetric rms Technology Materials delivery lag τ: D t = Z t τ

28 Details Continuum of symmetric rms Technology Materials delivery lag τ: D t = Z t τ Leontief technologies: Y Mt = minfd t /b M, N M g, Y Ft = minfx t /b F, N F g

29 Details Continuum of symmetric rms Technology Materials delivery lag τ: D t = Z t τ Leontief technologies: Y Mt = minfd t /b M, N M g, Y Ft = minfx t /b F, N F g Resource constraints: M t = M t 1 + Y Mt X t U t = U t 1 + O t Y Ft 1 M t 0 U t Y Ft

30 Details Continuum of symmetric rms Technology Materials delivery lag τ: D t = Z t τ Leontief technologies: Y Mt = minfd t /b M, N M g, Y Ft = minfx t /b F, N F g Resource constraints: M t = M t 1 + Y Mt X t U t = U t 1 + O t Y Ft 1 M t 0 U t Y Ft Stochastic R orders for rm i: O it = ρo t vit di = η t + v it. where

31 Firm s problem max E 0 β τ t [py F τ qd τ w (N M τ + N F τ )] τ=t subject to technology, uncertainty. For simplicity, take p, q, w as exogenous, xed.

32 Firm s problem max E 0 β τ t [py F τ qd τ w (N M τ + N F τ )] τ=t subject to technology, uncertainty. For simplicity, take p, q, w as exogenous, xed. For p su ciently large, rm always wants to sell as much as possible, i.e. Y Ft = min U t, b 1 F Mt 1 + bm 1 D t X t = min M t 1 + b 1 M D t, b F U t

33 Firm s problem max E 0 β τ t [py F τ qd τ w (N M τ + N F τ )] τ=t subject to technology, uncertainty. For simplicity, take p, q, w as exogenous, xed. For p su ciently large, rm always wants to sell as much as possible, i.e. Y Ft = min U t, b 1 F Mt 1 + bm 1 D t X t = min M t 1 + b 1 M D t, b F U t Then rm just optimizes over Z t, trade o carrying costs of M versus delayed sales due to un lled orders.

34 Firm s problem (cont.) Suppose τ = 1. (Note: τ = 0 implies M t = 0, Y Ft = O t.)

35 Firm s problem (cont.) Suppose τ = 1. (Note: τ = 0 implies M t = 0, Y Ft = O t.) Solution will take the form D t = Z t 1 = b M [b F (E t 1 fu t g + κ) M t 1 ] where κ depends on distribution of η t + v it, the markup, and the discount rate.

36 Firm s problem (cont.) Suppose τ = 1. (Note: τ = 0 implies M t = 0, Y Ft = O t.) Solution will take the form D t = Z t 1 = b M [b F (E t 1 fu t g + κ) M t 1 ] where κ depends on distribution of η t + v it, the markup, and the discount rate.

37 Firm s problem (cont.) Suppose τ = 1. (Note: τ = 0 implies M t = 0, Y Ft = O t.) Solution will take the form D t = Z t 1 = b M [b F (E t 1 fu t g + κ) M t 1 ] where κ depends on distribution of η t + v it, the markup, and the discount rate. This implies sales (total expenditures on nal goods) and production (sales plus the change in inventory) satisfy Y Ft = b 1 F X t = E t 1 fo t g + η t 1 + min fκ, η t g min κ, η t 1 Y Ft + M t = E t 1 fo t g + η t 1 + (1 b F ) min fκ, η t g min κ, η t 1

38 Firm s problem (cont.) Here is a slightly simpli ed (abstracting from labor costs and idiosyncratic risk) derivation. The rst-order condition for incrementally adding to orders Z t is c + p dy Ft+1 dy + Ft+1 βc 1 = 0 dz t dz t where c = qb M b F. Turns out dy Ft+1 = Pr (b F U t+1 > M t + D t+1 /b M ) dz t The probability is of a materials stockout at data t + 1.

39 Firm s problem (cont.) Here is a slightly simpli ed (abstracting from labor costs and idiosyncratic risk) derivation. The rst-order condition for incrementally adding to orders Z t is c + p dy Ft+1 dy + Ft+1 βc 1 = 0 dz t dz t where c = qb M b F. Turns out dy Ft+1 = Pr (b F U t+1 > M t + D t+1 /b M ) dz t The probability is of a materials stockout at data t + 1. So p Pr (b F U t+1 > M t + D t+1 /b M ) c + Suppose η t has a c.d.f. of G. We then get p c κ = b F G 1. p βc βc [1 Pr (b F U t+1 > M t + D t+1 /b M )

40 Firm s problem (cont.) Now suppose τ = 2.

41 Firm s problem (cont.) Now suppose τ = 2. Solution again will take the form D t = Z t 2 = b M bf E t 2 fu t g + κ 0 E t 2 fm t 1 g where κ 0 depends on distribution of η t 1 + v it 1 and η t + v it, the markup, and the discount rate.

42 Firm s problem (cont.) Now suppose τ = 2. Solution again will take the form D t = Z t 2 = b M bf E t 2 fu t g + κ 0 E t 2 fm t 1 g where κ 0 depends on distribution of η t 1 + v it 1 and η t + v it, the markup, and the discount rate.

43 Firm s problem (cont.) Now suppose τ = 2. Solution again will take the form D t = Z t 2 = b M bf E t 2 fu t g + κ 0 E t 2 fm t 1 g where κ 0 depends on distribution of η t 1 + v it 1 and η t + v it, the markup, and the discount rate. This implies sales (total expenditures on nal goods) and production (sales plus the change in inventory) satisfy Y Ft = E t 2 fo t g + (1 + ρ) η t 2 + min κ 0, η t + (1 + ρ) η t 1 Y Ft + M t = E t 2 fo t g + (1 + ρ) η t 2 + (1 b F ) min κ 0, η t + (1 + ρ) η t 1

44 Aggregation Suppose τ = 1. Let ξ t = Z Z κ η t σ min fκ, η t + σ v vg φ (v) dv κ ηt (η t + σ v v) φ (v) dv + κ 1 Φ σ v

45 Aggregation Suppose τ = 1. Let ξ t = Z Z κ η t σ min fκ, η t + σ v vg φ (v) dv κ ηt (η t + σ v v) φ (v) dv + κ 1 Φ σ v Aggregate M t is then M t = b F (κ ξ t )

46 Aggregation Suppose τ = 1. Let ξ t = Z Z κ η t σ min fκ, η t + σ v vg φ (v) dv κ ηt (η t + σ v v) φ (v) dv + κ 1 Φ σ v Aggregate M t is then M t = b F (κ ξ t ) Similarly, Y Ft = η t 1 + ρo t 1 + ξ t ξ t 1 Y Ft + M t = η t 1 + ρo t 1 + (1 b F ) (ξ t ξ t 1 )

47 Aggregation (cont.) We can linearize ξ around η t = 0 : κ κ ξ t t κ 1 Φ + Φ η σ v σ t = θη t v where κ θ Φ σ v Note that for nite κ, θ 2 (0, 1).

48 Aggregation (cont.) So we have Y Ft t (1 + ρ θ) η t 1 + ρ 2 O t 2 + θη t Y Ft + M t t (1 + ρ θ (1 b F )) η t 1 + ρ 2 O t 2 +θ (1 b F ) η t Var (Y Ft ) t (1 + ρ θ) 2 σ 2 η + ρ 4 σ 2 η/ 1 ρ 2 + θ 2 σ 2 η Var (Y t ) t (1 + ρ θ (1 b F )) 2 σ 2 η + ρ 4 σ 2 η/ 1 ρ 2 + θ 2 (1 b F ) 2 σ 2 η

49 Aggregation (cont.) So [Var (Y t ) Var (Y Ft )] /σ 2 η = (1 + ρ θ (1 b F )) 2 (1 + ρ θ) 2 + θ 2 (1 b F ) 2 θ 2 = 2θb F (ρ 2θ + θb F + 1) > 0 if 1 + ρ > (2 b F ) θ

50 Aggregation (cont.) So [Var (Y t ) Var (Y Ft )] /σ 2 η = (1 + ρ θ (1 b F )) 2 (1 + ρ θ) 2 + θ 2 (1 b F ) 2 θ 2 = 2θb F (ρ 2θ + θb F + 1) > 0 if 1 + ρ > (2 b F ) θ For su ciently large ρ, production is more volatile than sales. Note that ρ need not even be positive for this to be true.

51 Aggregation (cont.) Now let τ = 2. De ne ζ t Z min κ, η t + (1 + ρ) η t 1 + σ v v φ (v) dv κ 0 ηt t Φ + (1 + ρ) η σ t 1 v = θ 0 η t + (1 + ρ) η t 1 Y Ft = ρ 3 O t ρ + ρ 2 θ 0 (1 + ρ) η t 2 + θ 2 η t + ρη t 1 Y t = ρ 3 O t ρ + ρ 2 (1 b F ) θ 0 (1 + ρ) η t 2 + (1 b F ) θ 0 η t + ρη t 1

52 Aggregation (cont.) So Var (Y F ) /σ 2 η = ρ 6 / 1 ρ ρ + ρ 2 θ 0 (1 + ρ) 2 +θ 02 + θ 0 ρ 2 Var (Y ) /σ 2 η = ρ 6 / 1 ρ 2 + and 1 + ρ + ρ 2 (1 b F ) θ 0 (1 + ρ) 2 + (1 b F ) 2 θ 02 + (1 b F ) 2 θ 0 ρ 2 [Var (Y ) Var (Y F )] /σ 2 η = 2θ 0 b F ρ 2 + ρ + 1 ρ 2θ 0 + θ 0 b F + 1

53 Aggregation (cont.) So Var (Y F ) /σ 2 η = ρ 6 / 1 ρ ρ + ρ 2 θ 0 (1 + ρ) 2 +θ 02 + θ 0 ρ 2 Var (Y ) /σ 2 η = ρ 6 / 1 ρ 2 + and 1 + ρ + ρ 2 (1 b F ) θ 0 (1 + ρ) 2 + (1 b F ) 2 θ 02 + (1 b F ) 2 θ 0 ρ 2 [Var (Y ) Var (Y F )] /σ 2 η = 2θ 0 b F ρ 2 + ρ + 1 ρ 2θ 0 + θ 0 b F + 1 Compare 2θb F (ρ 2θ + θb F + 1) versus 2θ 0 b F ρ 2 + ρ + 1 ρ 2θ 0 + θ 0 b F + 1.

54 Simulations Standard deviations ρ = 0.9 θ =.7, θ 0 =.9 Y S Y/Y S/S τ = τ =

55 Conclusions The model seems capable of explaining

56 Conclusions The model seems capable of explaining The declines in both sales and output volatility

57 Conclusions The model seems capable of explaining The declines in both sales and output volatility The relatively larger decline in output volatility

58 Conclusions The model seems capable of explaining The declines in both sales and output volatility The relatively larger decline in output volatility The decline in inventory-sales ratio

59 Conclusions The model seems capable of explaining The declines in both sales and output volatility The relatively larger decline in output volatility The decline in inventory-sales ratio There is some direct evidence for the mechanism: Shorter lead times for materials orders

60 Conclusions The model seems capable of explaining The declines in both sales and output volatility The relatively larger decline in output volatility The decline in inventory-sales ratio There is some direct evidence for the mechanism: Shorter lead times for materials orders More work to be done on

61 Conclusions The model seems capable of explaining The declines in both sales and output volatility The relatively larger decline in output volatility The decline in inventory-sales ratio There is some direct evidence for the mechanism: Shorter lead times for materials orders More work to be done on Calibration, estimation

62 Conclusions The model seems capable of explaining The declines in both sales and output volatility The relatively larger decline in output volatility The decline in inventory-sales ratio There is some direct evidence for the mechanism: Shorter lead times for materials orders More work to be done on Calibration, estimation Dealing with the unit root aspect of the data to match evidence, model should look at growth rates