Credit Market Competition and Capital Regulation

Differentiating SW with respect to k , we have

dSW      (R - (1 - k)rD)rD           (1 - k)r_D²         (R - (1 - k)rD)rD

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

(1 - k)r_D²

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

2c

Calculating this expression at the two extreme levels of capital gives

and

r_D²

= 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 r_L =(1— k^CS)r_D + 2c, and banks
are required to hold capital k^cs equal to k^cs = min , 1j-. For k^cs = 1 (i.e., if c > rE),
banks earn profits Π = c — r_E > 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 8cr}E-RrD ⁺rD^-4 ∖ ""■' ^c(^4crΕ-R^rD ^+rD ) ₁ 1      . . . .

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

r_D

less than 1 for c > -R^ and equal to one otherwise. For k^cs = 1, Π = Rc — rE > 0, and
Π =0fork^cS < 1.

Proof: Start by noting that, since q = min ∣ ^rL (1_2c ^k)^rD , 11, if rL < (1 — k)rD + 2c then
q = ^rL-⁽1‘^cc^k)rD < 1. Since CS = q(R — rL), we have that ^d∂^cs = ∂k(R — rL) = rD(R — rL) > 0

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