Credit Market Competition and Capital Regulation

for q< 1. Therefore, more capital increases consumer surplus.

We proceed in two stages, starting by maximizing CS with respect to the loan’s price,
rL ,forafixed k , which yields

Solving the FOC yields r_L = R⁺(¹-^k)^rD.

We can now maximize CS with respect to the choice of capital, k. However, we know
from above that the combination of rL =   ' and the highest possible k will be

optimal for borrowers. We therefore introduce the participation constraint for the bank,
that Π = q(r_L — (1 — k)r_D) — krE — cq² ≥ 0. Substituting for q = ^rL~(2-k)^rD as well as for
r_L , we obtain

(R — (1 — k)r_D)²

∏ = (-----( _16c ) D) — krE ≥ 0, k ≤ 1.

We can solve this for the value of k that satisfies the constraint with equality (Π =0). Since
Π is strictly convex in k, 0 ≤ k ≤ 1, and consumer surplus is increasing in k, the relevant
solution must be either the smaller root or a corner solution at k =1. The solution is then

. 8 8crE — RrD + rD — 4 JrEc (4crE — RrD + rD )   |

mⁱⁿ ɪ----------------------rD----------------------■¹ ʃ •

Note that if k^CS = 1, then r_L = RR.

We now check when in fact q< 1. From the definition of the optimal level of monitoring

q = min I ^rL (¹2- ^k)^rD, 1}, we see that, for r_L ≥ (1 — k)rD + 2c, q = 1. Substituting in the

optimal value for rL gives the following condition:

The right hand side is maximized at k =0.Thus,asufficient condition for q =1is that

R — r_D — 4c ≥ 0. In this case, there is no benefit in terms of greater monitoring to having
27

<<   27   28   29   30   31   32   33   >>

More intriguing information