R k E 1 χ E Wt E n E t+1 t t + 1 n E t+1 = ( 1 χ E) W E t+1 c E t+1 = χ E Wt+1 E t + 1 q t K t Rt+1 E 1 Γ E t+1 Π t+1 = P t+1 /P t W E t+1 = ( 1 Γ E t+1 ) R E t+1q t K t Π t+1 Γ E t+1 K t q t q t K t j (, 1)
n E t b t = q t K t n E t t t + 1 ωt+1r E t+1 E Rt+1 E ωt+1 E iid E{ωt+1} E = 1 σt E = σ E ςt E ςt E ω E t+1 ω E Ft+1 E = F E (ω E t+1 t+1) = f E (ωt+1)dω E t+1 E f E ( ) t t+1 (1 δ) q t+1 t R E t+1 = rk t+1 + (1 δ) q t+1 q t Π t+1 ω E t+1 = Z tb t R E t+1q t K t = xe t R E t+1 x E t Z t b t /(q t K t ) ω E t+1 Rt+1 E ωt+1r E t+1q E t K t µ E
ω E t+1rt+1q E t K t (ωt+1 E ω E t+1)rt+1q E t K t Rt+1q E t K t Γ E t+1 = Γ E (ω E t+1) ω E t+1 ω E t+1f(ω E t+1)dω E t+1 + ( 1 F E t+1) ω E t+1 1 Γ E t+1 G E t+1 = G E (ω E t+1) ω E t+1 ω E t+1f E (ω E t+1)dω E t+1 {[ ( )] x E t 1 Γ E E t R Et+1q } Rt+1 E t K t R E t+1 n B t d t n B t + d t = b t. n B t ϕ t b t ϕ t R F t+1 = ( Γ E t+1 µ E G E t+1 ) R E t+1q t K t b t
E t { (1 Γ F t+1 ) [ Γ E ( x Ej t Rt+1 E ) µ E G E ( )] } x Ej t R Et+1q Rt+1 E t K t ϕ t E t { R B t+1 (q t K t n E t ) } 1 Γ F t+1 x E t K t+1 E t { Γ E t+1 + ξ t ( 1 Γ F t+1 ) ( Γ E t+1 µ E G E t+1) } = E t { ( 1 Γ E t+1) R E t+1 + ξ t [( 1 Γ F t+1 ) ( Γ E t+1 µ E G E t+1) R E t+1 ϕ t R B t+1] } = ξ t 1 χ B W B t n B t+1 t t + 1 n B t+1 = ( 1 χ B) W B t+1
c E t+1 = χ B W E t+1 R B t+1 W B t+1 = RB t+1n B t Π t+1. R B t+1 R B t+1 = ( ) 1 Γ F t+1 R F t+1 b t n B t 1 Γ F t+1 i ω F i t+1 i t + 1 Ξ F t+1 = max [ ω F t+1r F t+1b t R D t d t, ] ω F t+1 ω F t+1r F t+1b t = R D t d t R D t d t Ξ F t+1 = max [ ω F t+1 ω F t+1, ] R F t+1b t ω F t+1 E{ω F t+1} = 1 σ F t = σ F ς F t ς F t Ft F ω F Ft+1 F = F F (ω F t+1 t+1) f(ωt+1)dω F t+1 F G F t+1 = G F (ω F t+1) ω F t+1 ω F t+1f(ω F t+1)dω F t+1
µ F Rt+1b F t max [ ω F t+1 ω F t+1, ] = ω F t+1 = 1 Γ F t+1 ω F t+1f(ω F t+1)dω F t+1 ω F t+1 ω F t+1 f(ω F t+1)dω F t+1 Γ F t+1 = Γ F (ω F t+1) ω E t+1 ωt+1f(ω F t+1)dω F t+1 F + ( ) 1 Ft F ω F t+1 Ξ F t+1 = (1 Γ F t+1)r F t+1b t. Π ϕ τ R τ Π τ b R t R = ( Rt R ) τr ( Πt Π ) τπ ( ) τb bt b R t = R D t ϕ t ϕ = ( ) ζb bt b
max c tl td t E t [ ] β t+s ln c t+s φ l1+η t+s 1 + η s= < β < 1 c t l t φ η c t l t d t c t + d t + t t w t l t + RD t d t Π t + Ξ K t t t w t R D t t Π t = P t /P t Ξ K t w t = φl η t /Λ t 1 = E t { βt,t+1 R D t+1/π t+1 } βt,t+s = β t+s Λ t+s Λ t Λ t = 1/c t Y it i (, 1) P it Y t P t Y it P t Y t 1 Y itp it i Y t = ( 1 Y ε ε it i) ε ε ε > 1 i Yit d = (P it /P t ) ε Y t P t = ( 1 P 1 ε it di) 1 1 ε
P t = P it Y it = A t Kitl α 1 α it < α < 1 A t K it l it P it Y it P t r K t K it w t l it r K t w t l it = (1 α)s it Y it r K t K it = αs it Y it s it s t = w1 α t (rt K ) α 1 α α (1 α) 1 α i P it A t maxe t P it s= [ β t+s P it+s Yit+s d κ ( p P t+s 2 Π λp t P it P it 1 ) 2 ( Y it+s + s t+s Yit+s Yit+s) ] d κ p > λ p 1 Π t P it P t κ p Π t Π λp t ( Π t Π λp t 1 ) = εs t (ε 1) + κ p E t { β t,t+1 Π t+1 Π λp t ( Π t+1 κ p λ p = Π λp t 1 ) Y t+1 Y t } {I t } t= E t s= β t,t+s [q t+s x t+s I t+s ]
x t = K t (1 δ)k t = [ 1 κ I 2 δ κ I ( ) ] 2 It 1 I t I t ( 2 I t 2 I t 1) 1 = q t [ +E t { 1 κ I 2 ( ) 2 ( ) It It It 1 κ I 1 I t I t q t+1 β t,t+1 κ I ( It+1 I t 1 ) ( ) } 2 It+1 I t I t ] Y t = c t + χ E W E t + χ B W B t + I t + κ I 2 ( It I t 1 ) 2 I t + µ E G E t R E t q t K t Π t + µ F G F t Rt F b t Π t (1 α) s t Y t l t = φ tl η t Λ t R t ϕ t R t = Rt D {l t K t I t c t Y t n E t b t n B t d t x E t } t= {w t rt K q t Π t s t } t= {Rt F Rt E Rt B } t= {R t ϕ t } t= {A t ςt E ςt F } t= ln A t = ρ A ln A t + ε A t,
< ρ A < 1 ε A t iid σ A ωt+1 E ωt+1 F σt E = σ E ςt E ln ς E t ln ς F t = ρ E ln ς E t + ε E t = ρ F ln ς F t + ε F t < ρ E < 1 < ρ F < 1 σ E ς σ F ς iid ε E t ε F t ω E ω F µ E χ E σ E σ F F F A = ς E = ς F = 1 l = 1 φ β.99 R = 1 = 1.1.99 2% η = 1 1 1 φ =.749 α =.35 δ =.25 φ =.7624
ε = 6 ε ε = 1.2 κ p = 3 κ I = 2.43 R k Π = 1.5 2 ϱ qk n E 2 υ E = RE R D 2 F E =.75 3% 8% ϕ =.8 υ B = RB R D 6 µ F =.3 2% ω E =.498 µ E = 9% σ E =.271 χ E =.18 ω F =.92 σ F χ B =.25 =.14 1.36% µ F > µ E µ F
χ B.28.3 F F.4%.9% 2.7% R D R R F R E R F R B ϕ R B ln A t ln ς t 6 j=1 (xm j x d j) 2 x m j j x m j ρ A =.8638 ρ ς =.833 σ A =.716 σ E ς =.867
τ b = ζ b > ϕ τ b > ζ b = τ Π ζ b τ Π τ b (ζ b, τ Π ) ζ b = ζ b ζ b ζ b
τ b τ b τ π Π ϕ Π ϕ ζ b ϕ
Z t Z = ( Zt Z ) τr ( Πt Π ) τπ ( ) τb bt b R t = (Z t, 1) Z t R t ζ b [, 11] ζ b [12, 2] τ b [,.9] τ R = τ π =.9 τ π = 1.2
ω F ζ b ζ b 13 ζ b ζ b = 2 3% ζ b > ω F
ζ b = τ b τ b =
τ R ρ R = 1 { } W t = E t β s U (c t+s, 1 l t+s ), s= W t = U (c t, 1 l t ) + βe t {W t+1 }. { E t β [ s U ( )] } { (1 ω) c A t+s, 1 lt+s A = E t β [ s U ( c B t+s, 1 lt+s)] } B, s= s= ω 1 ζ b = τ b =.99
R D = Π β q = 1 R E = R D υ E R B = R D υ B r K = [ RE Π s = ε K = ε + κp [ ( 1 1 r K A α s (1 δ)]q ε (1 β) ( Π 1 λp 1 ) Π 1 λp ) ] 1 l α α I = δk ( ) Y = 1 r K α s K ( ) F E = 1 Φ ln ω E + 1 2(σ E ) 2 ω E σ E σ ( ) E G E ln ω = Φ E 2(σ 1 E ) 2 σ ( E ) G E = 1 Φ ln ω E 1 2(σ E ) 2 ω E σ E σ E Γ E = G E + ω E (1 F E ) Γ E = G E ( + (1 F E ) ) ω E F E G F ln ω = Φ F 1 2(σ F ) 2 σ F Γ F = ( 1 F F ) ω F + G F ξ = Γ E (1 Γ F )(Γ E µ E G E ) n E = (1 χ E ) ( 1 Γ E) R E qk Π b = qk n E R F = ϕ 1 Γ F R B n B = ϕb χ B = 1 β υ B c = Y χ E ( 1 Γ E) R E qk Π w = (1 α) s Y l φ = w cl η d = b n B x E = ω E R E χb RB n B Π µe G E RE qk Π µf G F RF b Π ω E µ E σ E χ E ω F σ F F F R D R E R B q r K s K I Y w G E Γ E G E F E Γ E ξ n E b n B d x E χ B R F G F Γ F c φ
= + 1 β (1 χe ) ( 1 Γ E) υ E ϱ E = Γ E + ξ E ( 1 Γ F ) ( Γ E µ E G E ) = = ( 1 Γ E) R E + ξ [( 1 Γ F ) ( ( ) Γ E µ E G E) R E ϕr B] = F E ln(ω + Φ E )+ 1 2(σ E ) 2 σ E = ω F + (1 ( ϕ) RD R F = F F + Φ ln(ω F )+ 1 2(σ F ) 2 σ F ) = R B + ( 1 Γ F ) 1 ϕ RF ω E µ E σ E χ E ω F σ F F F
β =.99 η = 1 φ =.7624 α =.35 δ =.25 ε = 6 κ p = 3 κ I = 2.43 σ A =.716 ρ A =.8638 σ ς =.867 ρ ς =.833 Π = 1.5
ϱ qk n E = 2 υ E RE R D = 1.5 4F E = 3 υ B RB R D = 1.15 ϕ =.8 µ F =.3 ω E =.498 χ E =.18 µ E =.9 σ E =.271 ω F =.92 σ F =.14 4F F = 1.36 χ B =.25 R = 1.151 R D = 1.151 R F = 1.164 R E = 1.22 R B = 1.34
τ R τ π τ b ζ b W ω ω
2 1.8 Multiple Unique Explosive Coefficient on inflation (τπ) 1.6 1.4 1.2 1.8.6.4.2 5 1 15 Coefficient on lending (ζ b ) τ b = ζ b > 2 1.8 Multiple Unique Explosive Coefficient on inflation (τπ) 1.6 1.4 1.2 1.8.6.4.2.2.4.6.8 1 1.2 1.4 1.6 1.8 2 Coefficient on lending (τ b ) τ b > ζ b =
ϕ = 12% 2 1.8 Multiple Unique Explosive Coefficient on inflation (τπ) 1.6 1.4 1.2 1.8.6.4.2 5 1 15 Coefficient on lending (ζ b ) τ b = ζ b > ϕ = 12% Monetary policy rate Output Investment Inflation.2.4.6.2.4.6.8.2 15 2 2 4 6 8 15 2 15 2.5.5 15 2 Return on capital 5 5 15 2 Entrep. net worth 5 5 2 15 2 Piecewise Linear 3 2 1 Bank net worth 15 2 Linear (ignores ZLB) Nominal loan growth 2 1.5 1.5 15 2 ζ b
1 Monetary policy rate.8.6 Ann.ppt.4.2 2 4 6 8 1 12 14 16 18 2 Quarters ζ =12 b ζ =16 b ζ =2 b τ π =.9 Monetary policy rate Output Investment Inflation.5.5 2.1.2.3.5.5 2 5 1 5 1 5 1 5 1 Return on capital.5.5 2 2.5 5 1 Entrep. net worth.5 2 2.5 3 5 1 Piecewise Linear Bank net worth 5 4 3 2 1 5 1 Linear (ignores ZLB) Nominal loan growth.5.4.3 5 1 ζ b
τ π =.9.5.5 Monetary policy rate 2.5.35.4.45.5.55 Output.5 Investment.2.4.6.8 2 2.2.5 Inflation.5.5 2 2.5.5 Return on capital Entrep. net worth 3.1.6 Bank net worth Nominal loan growth.7.5 2 2.5.5 3.2 3.3 3.4.5 Piecewise Linear.5.4.3.5 Linear (ignores ZLB).6.5.4.5 τ b
τ π = 2.5 ζ b τ π = 1.2 τ π = 2.5 Monetary policy rate Output Investment Inflation.4.2.5.6.7.8.9.1.2.3.4.2.2.4 15 2 15 2 15 2 15 2.2.4.6 Return on capital.1.2.3.4 Entrep. net worth 15 2 15 2 15 2 Piecewise Linear Linear (ignores ZLB) 8 7 6 Bank net worth Nominal loan growth.34.32.3.28.26 15 2 ζ b τ π =.75
τ π =.75.5 Monetary policy rate.1.2.3 Output.5.5 Investment.5.5 Inflation 5 1 5 1 5 1 5 1 Return on capital Entrep. net worth Bank net worth 5.5.5 4 2 3.5 2.5 2 3 1 5 1 5 1 5 1 Piecewise Linear Linear (ignores ZLB) Nominal loan growth.45.4.35.3 5 1 ζ b τ π =.75.5 Monetary policy rate.5.35.4.45.5 Output.55.5.4.6.8 Investment 2.5 Inflation.5.5.5 Return on capital.8.2.4.6.8 Entrep. net worth 3.1 3.2 3.3.6.5.4 Bank net worth Nominal loan growth.55.5.45.4.35.5.5 Piecewise Linear.5 Linear (ignores ZLB).5 τ b
Monetary policy rate Output Investment Inflation.5.5 2.1.2.3.5.5 2 5 1 15 5 1 15 5 1 15 5 1 15 Return on capital Entrep. net worth Bank net worth Nominal loan growth 6.5.5 4 2.4 2 2.5 2 3.3 5 1 15 5 1 15 5 1 15 5 1 15 Piecewise Linear Linear (ignores ZLB) ζ b R t R = ( Rt R ) τr ( Πt Π ) τπ ( ) τb bt /Y t b/y ϕ t ϕ = ( ) ζb bt /Y t b/y
.5.5 2 Monetary policy rate.4.6.8 Output.5 2 2.5 Investment 2 Inflation.5.5.5.5 Return on capital Entrep. net worth Bank net worth Nominal loan growth.5 2 2.5.5 3.2 3.4 3.6.5 Piecewise Linear.6.5.4.3.5 Linear (ignores ZLB) 2.5 2 1.5 1.5.5 τ b