Sticky Leverage. Joao Gomes, Urban Jermann & Lukas Schmid. Wharton School, Duke & UCLA

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1 Sticky Leverage Joao Gomes, Urban Jermann & Lukas Schmid Wharton School, Duke & UCLA

2 We consider nominal debt that is long-term and defaultable

3 We consider nominal debt that is long-term and defaultable Deflation risk for firms nominal debt

4 We consider nominal debt that is long-term and defaultable Deflation risk for firms nominal debt default

5 We consider nominal debt that is long-term and defaultable Deflation risk for firms nominal debt default debt overhang

6 We consider nominal debt that is long-term and defaultable Deflation risk for firms nominal debt default debt overhang We study this channel for monetary non-neutrality

7 Preview of findings Debt deflation is quantitatively powerful

8 Preview of findings Debt deflation is quantitatively powerful Sticky leverage is key

9 Preview of findings Debt deflation is quantitatively powerful Sticky leverage is key Taylor rules stabilize output

10 Related literature Debt deflation: De Fiore, Teles, and Tristani (2011), Kang and Pflueger (2012), Christiano, Motto and Rostagno (2009), Bhamra, Fisher and Kuehn (2011) Debt overhang: Occhino and Pescatori (2012), Moyen (2007), Hennessy (2004), Chen and Manso (2010) Default, general equilibrium: Gomes and Schmid (2013), Gourio (2012), Miao and Wang (2010) No quantitative business cycle analysis with nominal, long-term debt

11 Model Continuum of firms of measure one, firm j produces ) yt j = A t (k t j α ( ) nt j 1 α with aggregate productivity ln A t = ρ ln A t 1 + σε t, and ) ( ) k j t+1 (1 = δ + it j kt j g it j kt j

12 Model Continuum of firms of measure one, firm j produces ) yt j = A t (k t j α ( ) nt j 1 α with aggregate productivity ln A t = ρ ln A t 1 + σε t, and ) ( ) k j t+1 (1 = δ + it j kt j g it j kt j Define ) R t kt j max A t (k j α ( nt j t nt) j 1 α wt nt j

13 Model Continuum of firms of measure one, firm j produces ) yt j = A t (k t j α ( ) nt j 1 α with aggregate productivity ln A t = ρ ln A t 1 + σε t, and ) ( ) k j t+1 (1 = δ + it j kt j g it j kt j Define ) R t kt j max A t (k j α ( nt j t nt) j 1 α wt nt j After-tax operational profits, with idiosyncratic IID shock z j t ( ) (1 τ) R t kt j zt j kt j

14 Debt Nominal debt outstanding requires payment (c + λ) bj t µ t b t B t /P t 1, c coupon, λ amort., µ t inflation rate

15 Debt Nominal debt outstanding requires payment (c + λ) bj t µ t b t B t /P t 1, c coupon, λ amort., µ t inflation rate After issuing s j t p j t market value of debt b j t+1 = (1 λ) bj t µ t + sj t p j t

16 Default Firms default if (1 τ) ( R t k j t z j t k j t ) + V t (k j t, b j t, µ t ) < ((1 τ) c + λ) bj t µ t

17 Default Firms default if ( ) ) (1 τ) R t kt j zt j kt j + V t (k t j, bt, j µ t < ((1 τ) c + λ) bj t µ t ( ) V t kt j, bt, j µ t = max b j t+1,kj t+1 p j t ( ) b j t+1 (1 λ) bj t µ t It j + τδkt j + E t M t,t+1 ( ) 0, (1 τ) R t+1 k j t+1 zj t+1 kj t+1 max ((1 τ) c + λ) bj t+1 µ t+1 ) +V t+1 (k dφ (z t+1) j t+1, bj t+1, µ t+1

18 Debt pricing b j t+1 pj t = E t M t,t+1 z j z j, t+1 ] Φ(z j, t+1 [c ) + λ + (1 λ) p j b j t+1 t+1 µ + t+1 ( ) R (1 τ) t+1 k j t+1 z j t+1 kj t+1 ξk t+1 ) +V t+1 (k j t+1, bj t+1, µ t+1 + (1 λ) pj t+1 bj t+1 µ t+1 dφ (z t+1 )

19 Households, equilibrium max E {C,N} β t [(1 θ) ln C t + θ ln (3 N t )] t=0

20 Households, equilibrium max E {C,N} β t [(1 θ) ln C t + θ ln (3 N t )] t=0 Aggregate resource constraint y t [1 Φ (z )] ξ r ξk t = C t + I t

21 Households, equilibrium max E {C,N} β t [(1 θ) ln C t + θ ln (3 N t )] t=0 Aggregate resource constraint Inflation y t [1 Φ (z )] ξ r ξk t = C t + I t ln µ t = (1 ρ µ ) ln µ + ρ µ ln µ t 1 + ε µ t

22 Characterization Debt Overhang and Sticky Leverage v (ω, µ) = max ω,i z z [ ( ) p ω g (i) (1 λ) ω µ i + τδ +g (i) EM (1 τ) (R z ) ] ((1 τ) c + λ) ω µ + v (ω, µ ) dφ (z ) with ω b/k, i I /k, v V /k State of economy: (ω, K, µ, A)

23 Debt Overhang FOC i 1 pω = EM z z (1 τ) (R z ) ((1 τ) c + λ) ω µ +v (ω, µ ) dφ ( z )

24 Debt Overhang FOC i 1 pω = EM z z (1 τ) (R z ) ((1 τ) c + λ) ω µ +v (ω, µ ) dφ ( z ) Assume µ i.i.d., ξ r = 0, and no shocks for a long time so that µ t 1 = µ, ω t = ω

25 Debt Overhang FOC i 1 pω = EM z z (1 τ) (R z ) ((1 τ) c + λ) ω µ +v (ω, µ ) dφ ( z ) Assume µ i.i.d., ξ r = 0, and no shocks for a long time so that µ t 1 = µ, ω t = ω Then, shock on µ t has no effect on i if ω t+1 = ω

26 Sticky Leverage FOC ω pg (i) + p ( ω ω g (i) (1 λ) ω ) µ = g (i) EM Φ ( z ) 1 [ (1 τ) c + λ + (1 λ) p ] µ

27 Sticky Leverage Long-term debt, λ < 1 FOC ω pg (i) + p ( ω ω g (i) (1 λ) ω ) µ = g (i) EM Φ ( z ) 1 [ (1 τ) c + λ (1 λ) p ] µ

28 Sticky Leverage Long-term debt, λ < 1 FOC ω pg (i) + p ( ω ω g (i) (1 λ) ω ) µ = g (i) EM Φ ( z ) 1 [ (1 τ) c + λ (1 λ) p ] µ Assume µ i.i.d., ξ r = 0, and no shocks for a long time so that µ t 1 = µ, ω t = ω

29 Sticky Leverage Long-term debt, λ < 1 FOC ω pg (i) + p ( ω ω g (i) (1 λ) ω ) µ = g (i) EM Φ ( z ) 1 [ (1 τ) c + λ (1 λ) p ] µ Assume µ i.i.d., ξ r = 0, and no shocks for a long time so that µ t 1 = µ, ω t = ω A negative shock on µ t increases ω t+1 > ω.

30 Sticky Leverage Long-term debt, λ < 1 FOC ω pg (i) + p ( ω ω g (i) (1 λ) ω ) µ = g (i) EM Φ ( z ) 1 [ (1 τ) c + λ (1 λ) p ] µ Assume µ i.i.d., ξ r = 0, and no shocks for a long time so that µ t 1 = µ, ω t = ω A negative shock on µ t increases ω t+1 > ω. Sticky leverage: high ω/µ high ω B k /P

31 Calibration Parameter Description Value β Subjective Discount Factor 0.99 γ Risk Aversion 1 θ Elasticity of Labor 0.63 α Capital Share 0.36 δ Depreciation Rate λ Debt Amortization Rate 0.06 ξ r Fraction of Resource Cost 1 ξ Default Loss 0.38 τ Tax Wedge 0.40 η 1 Distribution Parameter

32 Inflation shock

33 Monetary policy rule ˆr f t = ρ R ˆr f t 1 + (1 ρ R ) { ν µ ˆµ t + ν y Ŷ t } + ζt

34 Monetary policy rule ˆr f t ˆr f t = ρ R ˆr f t 1 + (1 ρ R ) { ν µ ˆµ t + ν y Ŷ t } + ζt = 0.6 ˆr f t { 1.5 ˆµ t Ŷ t } + ζt

35 Productivity shock

36 Wealth/capital shock

37 Conclusion Model with nominal long-term debt produces strong inflation non-neutrality without sticky prices

38 Conclusion Model with nominal long-term debt produces strong inflation non-neutrality without sticky prices Key mechanisms: sticky leverage and debt overhang

39 Conclusion Model with nominal long-term debt produces strong inflation non-neutrality without sticky prices Key mechanisms: sticky leverage and debt overhang Taylor rule implies inflation in response to low productivity and capital loss

40 One-period debt, λ = 1 FOC ω p + p ω ω = EM Φ ( z ) [ ((1 τ) c + 1) 1 µ ]

41 One-period debt, λ = 1 FOC ω p + p ω ω = EM Φ ( z ) [ ((1 τ) c + 1) 1 µ ] Assume µ i.i.d., ξ r = 0, and no shocks for a long time so that µ t 1 = µ, ω t = ω

42 One-period debt, λ = 1 FOC ω p + p ω ω = EM Φ ( z ) [ ((1 τ) c + 1) 1 µ ] Assume µ i.i.d., ξ r = 0, and no shocks for a long time so that µ t 1 = µ, ω t = ω Then, shock on µ t has no effect on ω t+1 = ω

43 Idiosyncratic Shocks Use general quadratic approximation φ(z) = η 1 + η 2 z + η 3 z 2 Symmetry z = z = 1, and E (z) = 0

44 Idiosyncratic Shocks Use general quadratic approximation φ(z) = η 1 + η 2 z + η 3 z 2 Symmetry z = z = 1, and E (z) = 0 One free parameter η 1

45 Shocks [ at µ t ] [ ρa ρ = a,µ ρ a,µ ρ µ ] [ ] [ at 1 ε a + t µ t 1 ε µ t ]

46 Shocks [ at µ t ] [ ρa ρ = a,µ ρ a,µ ρ µ ] [ ] [ at 1 ε a + t µ t 1 ε µ t ] AR(1) calibration ρ a = 0.97, σ a = ρ µ = 0.85, σ µ = ρ a,µ = ρ a,µ = ρ µa = 0

47 Shocks [ at µ t ] [ ρa ρ = a,µ ρ a,µ ρ µ ] [ ] [ at 1 ε a + t µ t 1 ε µ t ] AR(1) calibration ρ a = 0.97, σ a = ρ µ = 0.85, σ µ = VAR calibration A = ρ a,µ = ρ a,µ = ρ µa = 0 [ ] σ a = , σ µ = , ρ µa = 0.19

48 Data Model Model AR(1) VAR(1) First Moments Investment/Output, I /Y Leverage, ω Default Rate, 1 Φ(z ) 0.42% 0.42% 0.42% Credit Spread 0.39% 0.39% 0.39% Second Moments σ Y 1.66% 1.58% 1.67% σ I /σ Y σ C /σ Y σ N /σ Y σ ω 1.7% 1.5% 1.7%

49 Variance decomposition, AR(1) Y Inv Cons Hrs Lev Default Bench., ω = 0.42 TFP shock a Inflation shock µ Low Lev, ω = 0.32 TFP shock a Inflation shock µ High Lev, ω = 0.52 TFP shock a Inflation shock µ

50 Variance decomposition, AR(1) Y Inv Cons Hrs Lev Default Bench., λ = 0.06 TFP shock a Inflation shock µ Long matu., λ = 0.03 TFP shock a Inflation shock µ One-period debt, λ = 1 TFP shock a Inflation shock µ

51 Monetary policy shock

52 I m advocating 6 percent inflation for at least a couple of years, says Rogoff, 56, who s now a professor at Harvard University. It would ameliorate the debt bomb and help us work through the deleveraging process. (Bloomberg May 19, :01 EDT).

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