Proof The result for c and f follows from Lemma 1 Take any other parameter or

Proof the result for c and f follows from lemma 1

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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
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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
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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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