Credit Market Competition and Capital Regulation



DD = DD) if the difference kreg - kcS is either concave or convex in rD. 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 ) )

1 (16cde - R2 ∕DEj)


> 0,


(4cde - RrD + rD ) /∕CceeRrD + rD}

R2
since by assumption c >    .

The finding that the function kreg - kCS is convex implies that kreg - kCS 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 of
rD, kreg-kCS < 0,
while for high values of
rD , kreg - kCS > 0. Therefore, kreg - kCS =0at one unique point,
which implies that D
D = 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 kcS = -c-. For the case with regulation, we have that for R > max{2c2rE-rD, Cc},
q
= 1 and kreg = 4c+rD-R. Therefore, kreg kcS

Cc + rD - R

rD       rE

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

R-Cc

DD < de ------ ,

rE - c

which establishes that kreg - kcS < 0 if and only if DD < DD = rE 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 qreq 1. This occurs for

31



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