Sticky Leverage João Gomes, Urban Jermann & Lukas Schmid Wharton School and UCLA/Duke September 28, 213
Introduction Models of monetary non-neutrality have traditionally emphasized the importance of sticky prices and/or wages This seems perhaps overdone We focus on an alternative channel for monetary non-neutrality Nominal debt that is both long-term and defaultable This is both large and quite costly to adjust (at least the principal) This creates two problems for firms Default risk Debt overhang
Preview of Findings Debt deflation is a quantitatively powerful propagation mechanism Sticky or persistent leverage is the key Conventional Taylor rules can stabilize output in response to shocks
Related Literature Debt deflation: De Fiore, Teles, and Tristani (211), Kang and Pflueger (212), Christiano, Motto and Rostagno (29), Bhamra, Fisher and Kuehn (211) Debt overhang: Occhino and Pescatori (212), Moyen (27), Hennessy (24), Chen and Manso (21) Default, general equilibrium: Gomes and Schmid (213), Gourio (212), Miao and Wang (21) No quantitative business cycle analysis with defaultable nominal, long-term debt
Model Continuum of firms of measure one, firm j produces y j t = A t (k j t ) α ( ) 1 α nt j 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 ) α ( 1 α R t kt j max A t (k j nt j t nt) j wt nt j After-tax operational profits, with idiosyncratic IID shock zt j ( ) (1 τ) R t kt j zt j kt j
Debt Nominal debt outstanding requires payment (c + λ) bj t µ t where 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
Equity Value and Default Value to equity holders/owners ) E (k t, j bt, j zt j, µ t = max where ) V (k t, j bt, j µ t Firms default when = max b j t+1,kj t+1 [, (1 τ) ( ) R t zt j kt j ((1 τ) c + λ) bj ( ) ] t + V k j µ t, bt, j µ t t { p j t ( ) b j t+1 (1 λ) bj t It j + τδkt j + µ t )} E t M t,t+1 E (k t, j bt, j zt j, µ t (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
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] bj t+1 µ t+1 + (R (1 τ) t+1 k j t+1 zj t+1 kj t+1 ( ) +V k j t+1, bj t+1, µ t+1 ξk t+1 + (1 λ) pj t+1 bj t+1 µ t+1 ) dφ (z t+1 )
Households and Equilibrium Consumer/Investor preferences max E {C,N} β t [(1 θ) ln C t + θ ln (3 N t )] t= Aggregate resource constraint Inflation Process 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 v (ω, µ) = max ω,i g (i) EM z z ( p [ ) i + τδ+ ω g (i) (1 λ) ω µ (1 τ) (R z ) ((1 τ) c + λ) ω µ + v (ω, µ ) ] dφ (z ) with ω b/k, v V /k State of economy: (ω, K, µ, A)
Optimal Leverage FOC for ω pg (i) + p ( ω ω g (i) (1 λ) ω ) µ = g (i) EM Φ ( z ) 1 µ [ (1 τ) c + λ + (1 λ) p ]
Sticky Leverage One-period debt, λ = 1 Proposition: p + p ω ω = EM Φ ( z ) [((1 τ) c + 1) 1µ ] Assume µ i.i.d., ξ r =, 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
Sticky Leverage Long-term debt, λ < 1 Proposition pg (i) + p ( ω ω g (i) (1 λ) ω ) µ = g (i) EM Φ ( z ) 1 [ ((1 τ) c + λ) p µ (1 λ) ] Assume µ i.i.d., ξ r =, 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
Optimal Investment and Debt Overhang FOC for i z 1 pω = EM z (1 τ) (R z ) ((1 τ) c + λ) ω µ +v (ω, µ ) dφ ( z ) Proposition: Assume µ i.i.d., ξ r =, and no shocks for a long time so that µ t 1 = µ, ω t = ω, R t+1 > R Then, shock on µ t has no effect on R (and i) iff ω t+1 = ω However if ω t+1 > ω then R t+1 > R
Calibration Parameter Description Value β Subjective Discount Factor.99 γ Risk Aversion 1 θ Elasticity of Labor.63 α Capital Share.36 δ Depreciation Rate.25
Calibration Parameter Description Value β Subjective Discount Factor.99 γ Risk Aversion 1 θ Elasticity of Labor.63 α Capital Share.36 δ Depreciation Rate.25 λ Debt Amortization Rate.6 τ Tax Wedge.4 η 1 Distribution Parameter.6617 ξ Default Loss.38 ξ r Fraction of Resource Cost 1
Idiosyncratic Shocks Use general quadratic approximation to p.d.f.: φ(z) = η 1 + η 2 z + η 3 z 2 Symmetry z = z = 1, and E (z) = One free parameter η 1
Shocks VAR process for inflation and productivity [ ] [ ] [ at ρa ρ = a,µ at 1 µ t ρ a,µ ρ µ µ t 1 ] + [ ε a t ε µ t ] Estimated values Γ = [.98.94.12.85 ] σ a =.74, σ µ =.45, ρ µa =.19 AR(1) version: ρ a =.97, σ a =.7 ρ µ =.85, σ µ =.4 ρ a,µ = ρ a,µ = ρ µa =
Inflation shock x 1-3 μ Y -2-4 -.5-6 2 4 -.1 2 4 I 5 x 1-3 c 2 x 1-3 r -.1 -.2 -.3-5 -2 -.4 2 4-1 2 4-4 2 4 2 x 1-3 N 2 x 1-3 defr.15 p ω -2 1.5 1.5.1.5-4 2 4 2 4 2 4
Key Moments Data Model Model AR(1) VAR(1) First Moments Investment/Output, I /Y.21.24.24 Leverage, ω.42.42.42 Default Rate, 1 Φ(z ).42%.42%.42% Credit Spread.39%.39%.39% Second Moments σ Y 1.7% 1.6% 1.7% σ I /σ Y 4.12 4.22 4.48 σ C /σ Y.54.41.43 σ N /σ Y 1.7.49.54 σ ω 1.7% 1.5% 1.7%
Variance decomposition, AR(1) Y Inv Cons Hrs Lev Default Benchmark, ω =.42 TFP shock a.63.37.39.6.1.5 Inflation shock µ.37.63.61.4.9.95 Low Leverage, ω =.32 TFP shock a.84.65.83.89.1.1 Inflation shock µ.16.35.17.11.9.99 High Leverage, ω =.52 TFP shock a.4.21.29.55.3.3 Inflation shock µ.6.79.71.45.97.97
Variance decomposition, AR(1) Y Inv Cons Hrs Lev Default Benchmark, λ =.6 TFP shock a.63.37.39.6.1.5 Inflation shock µ.37.63.61.4.9.95 Long maturity, λ =.3 TFP shock a.45.26.65.8.83.1 Inflation shock µ.55.74.35.2.17.99 One period debt, λ = 1 TFP shock a 1 1 1 1.1.2 Inflation shock µ.....99.98
Monetary policy rule Taylor rule with interest rate smoothing ˆr f t = ρ R ˆr f t 1 + (1 ρ R ) {ν m ˆµ t + ν y ŷ t } + ζ t Calibration ˆr f t =.6 ˆr f t 1 +.4 {1.5 ˆµ t +.5 ŷ t } + ζ t
Monetary policy shock 4 x 1-3 ζ 2 x 1-3 μ Y 3 2-2 -.5 1-4 2 4-6 2 4 -.1 2 4 I 5 x 1-3 c 2 x 1-3 rf -.1 -.2 -.3 -.4 2 4 Policy rule -5 Exogenous inflation -1 2 4-2 -4 2 4 2 x 1-3 N 2 x 1-3 defr.15 p ω -2 1.5 1.5.1.5-4 2 4 2 4 2 4
Productivity shock 8 x 1-3 a x 1-3 μ.15 Y 6 4 2-2 -4.1.5 Policy rule Exogenous inflation 2 4-6 2 4 2 4.4 I 8 x 1-3 c 1 x 1-3 rf.2 6 4 2-1 -2 -.2 2 4 6 x 1-3 N 4 2-2 2 4 2 4 2 x 1-3 defr 1-1 2 4-3 2 4 1 x 1-3 p ω 5-5 2 4
Wealth/capital shock 3 δ 15 x 1-3 μ.1 Y 2 1 1 5 -.1 -.2 2 4-5 2 4 -.3 2 4.3.2.1 -.1 2 4 I -.1 -.2 c Policy rule Exogenous inflation -.3 2 4 1 x 1-3 rf 5-5 2 4.3 N 1 x 1-3 defr.3 p ω.2.1 5.2.1 2 4-5 2 4 -.1 2 4
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 a significant increase in inflation in response to both low productivity and wealth shocks