Proof of Proposition 6: We begin with the case of parameter values such that q, k < 1,
and show that there exists a value reD > 0 such that kreg <kCS if and only if rD < reD .
Consider the solution that maximizes consumer surplus, kCS, and assume that R<4c and
c > 16R⅛, which implies that q, kCS < 1. From the condition defining kCS,
(R — (1 — kCS )rD ¢2 , ʌ
∏ = ʌ---⅛--— kCS ГЕ = 0,
16c
one can clearly see that, as rD → 0, kCS → 16R2^ < 1 for c > ∣(R2.
By contrast, kreg is defined by
∂SW
∂k
rD RR +(1 — kreg )rDλ
= 4c к 2 )
+ rD — rE =0.
For rD → 0, kreg → 0 as well, since it is optimal to just have deposit-based finance. These
two results together imply that there is some threshold rD such that, for rD < r D, kreg < kCS.
At the other extreme, we consider the solutions as rD → rE. For c > ι∣r~, kCS =
8c-R+rE-4√ (4c-R+rE )c
ГЕ
< 1.
By comparison, kreg → 1 as rD → rE for all parameter values.
Therefore, we can also conclude that there must exist some threshold rD such that, for
rD > rD, kreg > kCS.
Comparing the two values of k, kreg <kCS if and only if
kreg =
RrD + rD2 — 8c (rE — rD)
8crE — RrD + rD — 4 у/rEc (4сге — RrD + rD )
= kCS
Rearranging, we obtain the condition for kreg — kCS < 0 as:
ɪ RRrD + 4crD — 8сге + 2qcrE (4сге — RrD + rD)) < 0
Since we know that for low values of rD this condition will be satisfied, but not for higher
values, we can establish that there is a unique threshold where the inequality flips (i.e., that
30
More intriguing information
1. RETAIL SALES: DO THEY MEAN REDUCED EXPENDITURES? GERMAN GROCERY EVIDENCE2. PER UNIT COSTS TO OWN AND OPERATE FARM MACHINERY
3. Antidote Stocking at Hospitals in North Palestine
4. Evidence on the Determinants of Foreign Direct Investment: The Case of Three European Regions
5. The purpose of this paper is to report on the 2008 inaugural Equal Opportunities Conference held at the University of East Anglia, Norwich
6. Integration, Regional Specialization and Growth Differentials in EU Acceding Countries: Evidence from Hungary
7. Proceedings of the Fourth International Workshop on Epigenetic Robotics
8. AN IMPROVED 2D OPTICAL FLOW SENSOR FOR MOTION SEGMENTATION
9. Secondary stress in Brazilian Portuguese: the interplay between production and perception studies
10. The name is absent