Sticky Leverage Joao Gomes, Urban Jermann & Lukas Schmid Wharton School, Duke & UCLA
We consider nominal debt that is long-term and defaultable
We consider nominal debt that is long-term and defaultable Deflation risk for firms nominal debt
We consider nominal debt that is long-term and defaultable Deflation risk for firms nominal debt default
We consider nominal debt that is long-term and defaultable Deflation risk for firms nominal debt default debt overhang
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
Preview of findings Debt deflation is quantitatively powerful
Preview of findings Debt deflation is quantitatively powerful Sticky leverage is key
Preview of findings Debt deflation is quantitatively powerful Sticky leverage is key Taylor rules stabilize output
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
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
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
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
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
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
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
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
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 )
Households, equilibrium max E {C,N} β t [(1 θ) ln C t + θ ln (3 N t )] t=0
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
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
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)
Debt Overhang FOC i 1 pω = EM z z (1 τ) (R z ) ((1 τ) c + λ) ω µ +v (ω, µ ) dφ ( z )
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
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
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
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
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
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 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
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 0.025 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 0.6617
Inflation shock
Monetary policy rule ˆr f t = ρ R ˆr f t 1 + (1 ρ R ) { ν µ ˆµ t + ν y Ŷ t } + ζt
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 + 0.4 { 1.5 ˆµ t + 0.5 Ŷ t } + ζt
Productivity shock
Wealth/capital shock
Conclusion Model with nominal long-term debt produces strong inflation non-neutrality without sticky prices
Conclusion Model with nominal long-term debt produces strong inflation non-neutrality without sticky prices Key mechanisms: sticky leverage and debt overhang
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
One-period debt, λ = 1 FOC ω p + p ω ω = EM Φ ( z ) [ ((1 τ) c + 1) 1 µ ]
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
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
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
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
Shocks [ at µ t ] [ ρa ρ = a,µ ρ a,µ ρ µ ] [ ] [ at 1 ε a + t µ t 1 ε µ t ]
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 = 0.0070 ρ µ = 0.85, σ µ = 0.0040 ρ a,µ = ρ a,µ = ρ µa = 0
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 = 0.0070 ρ µ = 0.85, σ µ = 0.0040 VAR calibration A = ρ a,µ = ρ a,µ = ρ µa = 0 [ 0.98 0.094 0.012 0.85 ] σ a = 0.0074, σ µ = 0.0045, ρ µa = 0.19
Data Model Model AR(1) VAR(1) First Moments Investment/Output, I /Y 0.21 0.24 0.24 Leverage, ω 0.42 0.42 0.42 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 4.12 4.22 4.48 σ C /σ Y 0.54 0.41 0.43 σ N /σ Y 1.07 0.49 0.54 σ ω 1.7% 1.5% 1.7%
Variance decomposition, AR(1) Y Inv Cons Hrs Lev Default Bench., ω = 0.42 TFP shock a 0.63 0.37 0.39 0.60 0.10 0.05 Inflation shock µ 0.37 0.63 0.61 0.40 0.90 0.95 Low Lev, ω = 0.32 TFP shock a 0.84 0.65 0.83 0.89 0.10 0.01 Inflation shock µ 0.16 0.35 0.17 0.11 0.90 0.99 High Lev, ω = 0.52 TFP shock a 0.40 0.21 0.29 0.55 0.03 0.03 Inflation shock µ 0.60 0.79 0.71 0.45 0.97 0.97
Variance decomposition, AR(1) Y Inv Cons Hrs Lev Default Bench., λ = 0.06 TFP shock a 0.63 0.37 0.39 0.60 0.10 0.05 Inflation shock µ 0.37 0.63 0.61 0.40 0.90 0.95 Long matu., λ = 0.03 TFP shock a 0.45 0.26 0.65 0.80 0.83 0.01 Inflation shock µ 0.55 0.74 0.35 0.20 0.17 0.99 One-period debt, λ = 1 TFP shock a 1 1 1 1 0.01 0.02 Inflation shock µ 0.00 0.00 0.00 0.00 0.99 0.98
Monetary policy shock
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, 2009 00:01 EDT).