Credit Market Competition and Capital Regulation

Proof of Proposition 6: We begin with the case of parameter values such that q, k < 1,

and show that there exists a value reD > 0 such that k^reg <k^CS if and only if rD < reD .
Consider the solution that maximizes consumer surplus, k^CS, and assume that R<4c and
c > 16R⅛, which implies that q, kCS < 1. From the condition defining kCS,

(R — (1 — kCS )rD ¢2    , ʌ

∏ = ʌ---⅛--— kCS ГЕ = 0,

16c

one can clearly see that, as rD → 0, kCS → 16R2^ < 1 for c > ∣(R2.
By contrast, k^reg is defined by

rD RR +(1 — k^reg )rDλ
= 4c к 2       )

For r_D → 0, k^reg → 0 as well, since it is optimal to just have deposit-based finance. These
two results together imply that there is some threshold rD such that, for rD < r D, k^reg < kCS.

At the other extreme, we consider the solutions as rD → r_E. For c > ι∣r~, kCS =

By comparison, k^reg → 1 as rD → rE for all parameter values.

Therefore, we can also conclude that there must exist some threshold rD such that, for
rD > rD, k^reg > kCS.

Comparing the two values of k, k^reg <k^CS if and only if

Rearranging, we obtain the condition for k^reg — k^CS < 0 as:

ɪ RRrD + 4crD — 8сге + 2qcrE (4сге — RrD + rD)) < 0

Since we know that for low values of r_D this condition will be satisfied, but not for higher
values, we can establish that there is a unique threshold where the inequality flips (i.e., that

30

<<   30   31   32   33   34   35   36   >>

More intriguing information