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 16ckrD -16crD +R2
1 18crD — 48crD rE + 32crD r2E + 3R2γd rE — 2R2r2E
2 crD (4ге — 3rD)2
Comparing these two solutions, we note that as rD → 0, we have that, as before, kCS →
16Rr^. By contrast, for the case of social welfare maximization it is easy to see from the
FOC that as rD → 0, kreg → 0 as well. Therefore, for rD sufficiently small, we have that
kreg < 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
9 dq
Ге — 2cq^r
∂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 → rE. Substituting in for q, we can find
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