Proof.
The result for
c
and
f
follows from Lemma
1
.
Take any other parameter (or the
function
q
(
p, λ, θ
)) and call it Ξ. If
E
(
p

Ξ)
>
0 then the mapping
M
(
E

Ξ) is decreasing in
Ξ and increasing
E
. Therefore, we have
E
(
p

Ξ) =
M
(
E
(
p

Ξ)

Ξ)
>
M
(
E
(
p

Ξ)

Ξ
0
)
≥ M
2
(
E
(
p

Ξ)

Ξ
0
)
≥ M
n>
2
(
E
(
p

Ξ)

Ξ
0
)
≥ M
∞
(
E
(
p

Ξ)

Ξ
0
)
=
E
(
p

Ξ
0
)
,
which proves the result.
B
Industry Equilibrium
We first establish the existence of an industry equilibrium (Theorem
2
).
We then derive
conditions under which there is a unique rate of creative destruction (Proposition
2
).
To establish the existence of an equilibrium, we make the following assumption:
Assumption 1.
For the firm value, the order of the limit with respect to
f
and the supremum
over
c
can be interchanged:
lim
f
0
→
f
sup
c
{
E
(
p, c

f
0
) + (1

ξ
)
D
(
p, c

f
0
)
}
= sup
c
lim
f
0
→
f
{
E
(
p, c

f
0
) + (1

ξ
)
D
(
p, c

f
0
)
}
.
Theorem
2
(Equilibrium Existence)
.
If Assumption
1
holds then there exists an industry
equilibrium
Ψ
*
.
Proof.
The proof has several steps:
41
1. The first step is showing that the equity value converges to zero when
f
→ ∞
. Assume
this is not the case then for some
p
we have that
E
(
p

f
)
>
0 when
f
→ ∞
.
From
equation (
3
) it follows that for any
p >
0 with
E
(
p

f
)
>
0
0 =

rE
(
p

f
) + (1

π
)(
p

c
)
f
+
max
(
λ,θ
)
λ
(
E
θ
[
E
(min
{
p
+
x,
¯
p
}
)]

E
(
p

f
)
)

(1

π
)
q
(
p, λ, θ
)
f
+
p
{
E
(
p

1

f
)

E
(
p

f
)
}
.
Given that
E
(
p

f
)
≤
¯
p/r
and
λ
≤
¯
λ
, taking
f
→ ∞
implies that
0 =
p
{
E
(
p

1

f
=
∞
)

E
(
p

f
=
∞
)
}
and therefore that
E
(
p

f
=
∞
) =
E
(
p

1

f
=
∞
)
for any
p
for which
E
(
p

f
=
∞
)
>
0. Given that
E
(0

f
=
∞
) = 0 this implies that
E
(
p

f
=
∞
) = 0
,
which is a contradiction. Therefore, the equity value does converge to zero.
2. The debt value also goes to zero when
f
→ ∞
since the default time and the recovery
value in default go to zero. Therefore, firm value
V
(
f, θ
) and also entrant value
E
e
(
f
)
goes to zero as
f
→ ∞
.
3. Define firm value as
F
(
p
0

f, c
) =
E
(
p
0

f, c
) + (1

ξ
)
D
(
p
0

f, c
)
.
4. By Lemma
1
, equity value is continuous in
f
and therefore
lim
f
0
→
f
k
E
(
p

f, c
)

E
(
p

f
0
, c
)
k
= 0
.
As a result, the dynamics of
P
t
will also be the same under
f
and
f
0
→
f
. If in addition
the default threshold is the same then
lim
f
0
→
f
k
D
(
p

f, c
)

D
(
p

f
0
, c
)
k
= 0
since the default times will converge. Since the equity value is continuous in
f
, if the
default threshold is not the same then at
f
shareholders must be exactly indifferent
between default and no default. Take an arbitrary small
, because the equity value is
decreasing in
c
, for either
c

or
c
+
the default threshold under
f
0
→
f
will be the
same as the default threshold under
f
(and
c
). Furthermore, because the equity value
42
is continuous in
f
and
c
the dynamics of
P
t
will be continuous in both as well. This
implies that
lim
→
0
lim
f
0
→
f
k
D
(
p

f, c
)

D
(
p

f
0
, c
±
)
k
= 0
since the default time will converge. This implies that
lim
→
0
lim
f
0
→
f

F
(
p
0

f, c
)

F
(
p
0

f
0
, c
±
)

= 0
,
5. The previous step shows that for a given
f
,
c
, and
f
0
→
f
there exists an
c
0
= lim
→
0
c
±
such that the firm value is continuous in
f
. This implies that
sup
c
F
(
p
0

f
0
, c
) = sup
c
lim
f
0
→
f
F
(
p
0

f
0
, c
) = lim
f
0
→
f
sup
c
F
(
p
0

f
0
, c
)
.
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 Spring '17
 Enkhjin
 Debt, Test, The Natural