Proof of Proposition 4

9.1 Proof of Proposition 4

We begin by characterizing what the function b ∗ c (b) must look like in a well-behaved equilibrium. Using Proposition 3, we can write the equilibrium value function as:

⎨ u(τ ) + A ln g +

B(τ,g,b,b)

+ δEv c (b, A 0 ) ⎬ ⎪

⎪ ⎪ ⎪ max (τ,g) if A > b A(b) ⎪ ⎪ ⎪

: B(τ, g, b, b) ≥0

⎪ B(τ,g,b 0 ⎨ ⎪ ⎪ u(τ ) + A ln g + ,b)

v c (b, A) =

⎪ + δEv c (b 0 ,A 0 ) ⎪ ⎬

c (b)), b A(b)] ⎪

⎪ ⎪ ⎪ max (τ,g,b 0 )

if A

∈ [A ∗ (b, b

⎪ ⎪ : B(τ, g, b ⎪ 0 ⎩ ⎪ ⎪ , b) ≥0

+ δEv c (b p ∗ c (b), A 0 ) if A < A ∗ (b, b ∗ c (b)), where b A(b) is the threshold (possibly larger than A) such that for A ≥b A(b) the BBR constraint

B(τ ∗ , ⎩ qA ⎪ u(τ ∗ ) + A ln qA +

p ,b ∗ c (b),b)

that the debt level be less than b will bind. In this top range, the initial debt level will directly determine the debt level chosen next period. Using this and the assumption that the equilibrium is well-behaved, we have that

∂v c (b,A 0 ⎪ ) ⎪

if A > b A(b) ⎪

− 1 −τ b (A)(1+ε) n + δE ∂b

1 −τ c (b,A)

if A ∈ [A (b, b c (b)), b A(b)] (22)

∂b

⎪ ⎪ ⎪ ⎪ ³ db ∗ (b)/db

1 −τ c (b,A)(1+ε)

c n ∗ ∂b db (b)). Taking expectations, we obtain

∂v (b ∗ (b),A 0 ) db ∗ ⎩ (b) ⎪ c −(1+ρ) + δE c c c if A < A ∗ (b, b

∂v (b ∗ (b),A) db ∗ −δnE (b)

∂v c (b,A)

= G(A (b, b (b)))δ[1 + ρ

R min{A, b A(b)} 1 (b,A)

A ∗ (b,b ∗ c )

( 1 −τ c (b,A)(1+ε) )dG(A)

R A 1 (b,A)

+ min{A, b (

A(b)} 1 −τ b (A)(1+ε) )dG(A)(1 − δ) − δ nE c ∂b (1 − G(min{A, b A(b)})). (23) Now let b 0 and b 1 be as defined in (14) and (15). As explained in the text, we must have that

−τ b (A)

2 ∂v

b 0 is less than b 1 . We now characterize the end point b 0 . When b < b 0 , we know that b ∗ c (b) = b

db c ∗ and hence that (b)

db = 1. Moreover, since b ∗ c (b) = b, we have that b A(b) = A ∗ (b, b) and so we can rewrite (23) as:

∂v c (b, A)

−δnE dG(A). (24)

A ∗ (b,b)

1 −τ b (A)(1 + ε)

Since b q 0 + δEv c (b 0 ,A 0 ) is constant on the interval [b 0 ,b 1 ], the right hand derivative of the value function at b 0 and the left hand derivative at b 1 must be 1/q. Since the expected value function is differentiable, then, we must have that at b 0 the left hand side derivative (which is given by (24)) is 1/q implying that

dG(A) = .

A ∗ (b 0 ,b 0 )

1 −τ b 0 (A)(1 + ε)

This is (18). The next step is to characterize b ∗ (b) on the interval [b 0 ,b 1 ]. If b + δEv c (b, A c 0 q ) is constant on the interval [b 0 ,b 1 ] we must have that

∂v c for any b (b,A)

0 ,b 1 ]. Since E is a function of b ∈ [b ∗ ∂b c (b) and its derivative, (25) implies a differential equation that needs to be satisfied by b ∗ c (b) along with the initial condition b ∗ c (b 0 )=b 0 . Using (23), we can show that this condition requires that b ∗ c (b) in [b 0 ,b 1 ] is equal to a function

f (b) that solves the differential equation:

h df (b) ³

´i

q = G(A ∗ (b, f (b))) 1 − db δ 1 − n q

q )G(A − δ)G(A ∗ (b, f (b))) +

(b, b))

−( n

A ∗ (b,b) 1 −τ b (A)(1+ε) dG(A)(1 − δ) + δ q , with the initial condition f (b 0 )=b 0 . This is (19). Note that if this condition is satisfied, then any point in [b 0 , b] would be a legitimate choice for b ∗ c (b) when b ∈ [b 0 ,b 1 ]. We are therefore free to choose b ∗ c (b) as we like - in particular, b ∗ c (b) = f (b). By Theorem 2 0 in Braun (1992) (p.77),

−τ b n

f (b) is uniquely defined on [b 0 , b].

The final step is to pin down the end point of the interval b 1 . Because b c ∗ (b) is non decreasing and bounded in [b 0 ,b 1 ], it must be constant and equal to f (b 1 ) for debt levels b larger than b 1 .

db Using (23) and the fact that ∗ c (b)

db = 0 for b greater than b 1 we have that:

R min{A, b A(b)} ³

G(A (b, b (b))) +

1 −τ c (b,A)

c A ∗ (b,b c ∗ (b))

1 −τ c (b,A)(1+ε) dG(A)

1 −τ b (A)

∂v c (b, A)

min{A, b A(b)} 1 −τ b (A)(1+ε) dG(A)(1 − δ)

−δnE

. (26) ∂b

1 − δ(1 − G(min{A, b A(b)}))

for b greater than b 1 . The same logic used to pin down b 0 can now be used for b 1 : at b 1 we need the right hand side derivative (given by (26)) equal to 1/q. This implies that b 1 must satisfy:

1 b (A) (1

−δ) = 1 (1 −δ)G(A (b 1 ,b − −1 G(A ∗ (b

dG(A)(1 q

1 , f (b 1 )))+

A ∗ (b 1 ,b 1 )

1 −τ b 1 (A)(1 + ε)

This is (20). We now have a full characterization of the b ∗ c (b) function in a well-behaved equilibrium. Notice that for a given b ∗ c (b) function, (13) is a contraction with a unique fixed-point. Thus, since b ∗ c (b) is uniquely defined, there exists a unique well-behaved equilibrium.