Credit Market Competition and Capital Regulation



A Proofs

Proof of Proposition 1: Substituting q in bank profits, we obtain

∏ = (rL - (1 - k)rD)2 - krE.

4c

Differentiating profits with respect to k gives

dΠ      (rL - (1 - k)rD)rD

dk =        2C        rE

= qrD - rE 0,asq 1 and rE rD.

This implies that k = 0. Furthermore, for rL rD ,

dΠ
drL


(rL - (1 - k)rD)0 rL R,

2c                L ,

implying that bank profits are always increasing in the interest rate rL . However, since the
bank must satisfy the borrower’s participation constraint, the maximum interest rate that
can be charged satisfies
qc(R - rL) rB , where qc is the level of monitoring that borrowers
conjecture will take place. Since in equilibrium borrowers anticipate correctly how much
monitoring takes place, we have that
rL = R rB.

Proof of Proposition 2: Substituting rL = R rB and k = 0 in the expression for q gives

q =min


R rD
2c


,1 .


Thus, q =1if R rD 2c,andq< 1 if R rD 2c.

Substituting rL = R rB and keeping k > 0, social welfare becomes

SW = (R - (1 - k)rD)2 krE [1 (R - (1 - k)rD) ](1 k)rD.

4c                              2c

25



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