Credit Market Competition and Capital Regulation

To maximize social welfare in the absence of deposit insurance, we need to solve

max SW = Π + CS = qR - (1 - k) rD - krE - cq².
k

The FOC yields

∂SW  ∂q

-ɪ- = W (R — 2cq) + γd - ге = 0.
∂k    ∂k

Using ∂^dq = I     rD       , we can substitute into the FOC and solve for k, which yields

^∂k      16ckr_D -16cr_D +R²

1 18crD — 48crD rE + 32crD r²E + 3R²γ_d rE — 2R²r²E

²                crD (4ге — 3rD)²

Comparing these two solutions, we note that as r_D → 0, we have that, as before, k^CS →
₁₆Rr^. By contrast, for the case of social welfare maximization it is easy to see from the
FOC that as r_D → 0, k^reg → 0 as well. Therefore, for r_D sufficiently small, we have that
k^reg < kCS, as desired. □

Proof of Proposition 8: Note that bank profit maximization with respect to k yields the
following FOC:

^dq (^rL — ^{(1 - k})cD) + qcD ^- q(1— ^k)dcD
∂k                            ∂k

qcD — Ге — q(1 — k) ɪ.

∂k

Since qcD = γd, ^dCqD = — ^rD, and, given q = ɪ RR + RR² — 8cγd (1 — k) J when γl = R— ^rB,

that ^dq = _l ^rD      > 0, the FOC becomes :

^dk    _λ∕R²-8crD (1-k)     ’

rD         rD

ΓD — Ге + (1 — k)--.                  = 0.

q pR² — 8crD (1 — k)

Evaluating at k = 0, ∂∏ is clearly positive for rD → r_E. Substituting in for q, we can find

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