Credit Market Competition and Capital Regulation

DD = DD) if the difference k^reg - k^cS is either concave or convex in r_D. For this, we only
need the second derivative of the term inside the parenthesis, which yields

∂D₂ RDD _d + CcrD - 8cde + 2qcde (Ccde - RrD + rD ) )

(4cde - RrD + rD ) /∕Cce_e — RrD + rD}

R²
since by assumption c >    .

The finding that the function k^reg - k^CS is convex implies that k^reg - k^CS can at most cross
zero twice, the first time from above and the second from below. However, two crossings are
inconsistent with the finding in the proposition above that for low values ofrD, k^reg-k^CS < 0,
while for high values of r_D , k^reg - k^CS > 0. Therefore, k^reg - k^CS =0at one unique point,
which implies that DD = DD = eD , and we have just one threshold, as desired.

We next proceed to the case where q =1in both cases, which is true for sufficiently large
R, but that k< 1. Start with the case of consumer surplus maximization, where, for R> Cc,
q = 1 and k^cS = -c-. For the case with regulation, we have that for R > max{2c^2rE^-rD, Cc},
q = 1 and k^reg = ^4c+^rD^-R. Therefore, k^reg < k^cS ⇔

Cc + r_D - R

rD       rE

This last inequality can be solved for rD to yield the condition

R-Cc

DD < de ------ ,

rE - c

which establishes that k^reg - k^cS < 0 if and only if DD < DD = r_E R---c), as desired.

The last case is a possible “mixed” case, in which monitoring may be at a maximum
for one solution but not the other. It is straightforward to show that the only case of
relevance is where, with a slight abuse of notation, q^cS =1 but q^req < 1. This occurs for

31

<<   <   31   32   33   34   35   36   37   >   >>