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