Credit Market Competition and Capital Regulation

We know that, for q < 1, the optimal level of monitoring is q = ^rL (12- ^k)^rD. Substituting in
the value of r_L above we get q = R (14_ck)^rD. We therefore have that

^rD RR +(1 ^- k)^rD∖

4Ц-----2-----+^{+ r}D ^{- r}E = 0.

The solution we obtain is

_re         RrD + r_D² - 8c (rE - rD)

k ^g = min < -------D—2— -------,, 1 > .

rD²

From this expression, we obtain that k^reg < 1 for c > 8(_rRrD,_c) ⇔ R < 8c (^rE_r-^rD). Otherwise,
for R > 8c (^rE_r^-^rD), we have that kr^eg = 1.

The previous solution assumed that q< 1. To get the bounds on when q =1, substitute

the solution for k^reg, assuming k^reg < 1, into

R — (1 — kr^eg)rD   RrD — 4c (rE — rD)

4c                 2crD

From here, we see that for c > ↑_rR'2_r ⇔ R < 2c2^rE_r^-rD, q < 1. Otherwise, for R >
2c2rE_r^-rD, q = 1 and k should be set such that q(k) = 1 ⇔ k^reg = ^4c+rD R. Note, however,
that for R<4c this solution would imply that k^reg > 1, which is not feasible. Therefore, for

R < 4c, we obtain that k^reg = 1, which implies that q = R < 1.

One final point that needs to be verified is that, for R < 2c2^rE_r^-rD, then q = R~(1-k)^rD <

1, but that for R > 8c (^rE_r^^-rD ), we have that k^reg = 1, which would imply that q = R.

Note, however, that for both of these conditions to be true at the same time requires that

8c(rE-RD) < 2c2^rE_r^-rD. This will be satisfied if and only if 4 (rE — rD) < 2rE — rD ⇔
rE < 3rD. We can now use this in the necessary condition for q < 1, which is R < 2c2^rE_r^-rD.
Given the restriction on rE and rD, the right hand side must be less than 2c2(² r_rD) rD = 4c.
Therefore, the joint assumption that R < 2c^2rEr-^rD and R > 8c (^rE_r^^-rD) implies that R < 4c,

and consequently that q = R < 1, as desired. □

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