Credit Market Competition and Capital Regulation



Differentiating SW with respect to k , we have

dSW      (R - (1 - k)rD)rD           (1 - k)rD2         (R - (1 - k)rD)rD

^kuT =        2C         rE- [    2c     rD+       2c      ]=0

(1 - k)rD2

= ------D + rD - rE = 0.

2c

Calculating this expression at the two extreme levels of capital gives

dSW
dk


= rD - rE 0,
k=1


and

dSW
dk


rD2

= Dr + rD - rE R o,
k=0    2c

implying that the welfare-maximizing level of capital is k* (0,1) if rD > ʌ/e(e + 2rE) c,

and is given by

k* = 1 2c (rE rD ) < 1,
rD2

thus establishing the proposition.

Proposition 4B When there is an excess supply of funds, maximizing borrower surplus
yields the following equilibrium:

1) For R 4c, monitoring is q =1. The loan rate is rL =(1kCS)rD + 2c, and banks
are
required to hold capital kcs equal to kcs = min , 1j-. For kcs = 1 (i.e., if c > rE),
banks earn profits Π = c rE > 0, otherwise Π =0.

2) For R < 4c, monitoring is q = R-(1-k—)rD 1. The loan rate is rL = R+(1-k——,
C c c c c c C c CS , CS        ∙   8 8crE-RrD +rD-4 ""■' c(4crΕ-RrD +rD ) 1 1      . . . .

and banks hold capital equal to k = min < --------------⅝— -----------'-, 1 >, which is

rD

less than 1 for c > -R^ and equal to one otherwise. For kcs = 1, Π = Rc rE 0, and
Π =0forkcS < 1.

Proof: Start by noting that, since q = min ∣ rL (12c k)rD , 11, if rL (1 k)rD + 2c then
q = rL-(1‘cck)rD < 1. Since CS = q(R rL), we have that dcs = ∂k(R rL) = rD(R rL) > 0

26



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