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

= qr_D - r_E ≤ 0,asq ≤ 1 and r_E ≥ r_D.

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

(^rL ^- (1 ^{- k})^rD) ≥ 0 ∀rL ≤ R,

2c                ^L ,

implying that bank profits are always increasing in the interest rate r_L . However, since the
bank must satisfy the borrower’s participation constraint, the maximum interest rate that
can be charged satisfies q^c(R - rL) ≥ rB , where q^c 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 r_L = R — ^rB. □

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

Thus, q =1if R — r_D ≥ 2c,andq< 1 if R — r_D < 2c.

Substituting r_L = R — ^rB and keeping k > 0, social welfare becomes

SW = (^{R -} (^{1 - k})^rD)2 — kr_E — [1 — (^{R -} (^{1 - k})rD) ](1 — k)rD.

4c                              2c

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