Techical Appedix o Hamig eposis ad Helpig Bowes: The ispaae Impac of Ba Cosolidaio (o o be published bu o be made available upo eques. eails of Poofs of Poposiios 1 ad To deive Poposiio 1 s exac ad sufficie codiios f a geae umbe of Bs o educe he eail loa aes of small bas i mae we ca diecly diffeeiae equaio (6 aleaively diffeeiae equaio ( f he case of a small ba i mae : i im / 1 1 = E c m 1 i To fid 1 / we diffeeiae 1 i equaio (5 o obai / m / / m / m / / 1 1 1 = β / βm / β / βm / ( ( (T.1 β β β β β β Λ ( (T. Subsiuig i f 1 / ad 1 fom (T. ad equaio (5 (T.1 becomes im / 1 1 ( β ( i Λ = β β β Λ ( β im / β β Λ ( β im / / β Re-aagig he igh-had side of (T.3 oe fids ha i is egaive whe (T.3 1 1 im / β β β 0 >( m β β i β im / β β Λ ( β i (T.4
im / β β β 1 1 β βm i ( m β β Λ> (T.5 im / β β β i Recall ha β = ( - > 0 ad β / = - / < 0. By subsiuig i f fom equaio (4 i he pape i is saighfwad o show ha (β β / - β β / > 0 f > m. This alog wih he fac ha i / > 0 idicaes ha he umea of he em i baces is posiive. I addiio oe ca see ha he deomia is posiive so ha he aio i baces is posiive. ow by e-wiig his aio (T.5 ca be e-wie as Λ> β β 1 1 Ψ im im / ( m β β Ψim im / β β whee Ψim β im / ( β β < 0 he im / < 0 which pemis us o show ha m ( / 1 im β β im / β Ψ im (T.6 >0. I (T.6 sice β / Ψ / > 0. Thus a sufficie codiio f / < 0 is Λ>0. Similaly o fid he exac ad sufficie codiios f a geae umbe of Bs o educe he eail loa aes of small bas i mae we ca diffeeiae equaio ( f he case of a small ba i mae : i i / 1 1 = E c 1 i i (T.7 Subsiuig i f 1 / ad 1 fom (T. ad equaio (5 (T.7 becomes
i / 1 1 ( β ( i Λ = β β This deivaive is egaive whe β Λ ( β i / β β Λ ( β i / / β 1 1 i β β 0 > ( m β β β β i / β β Λ ( β i / i (T.8 (T.9 i / β β β 1 1 β βm i ( m β β Λ> i / β β β i which ca also be wie as (T.10 β β Ψ Λ> 1 1 i i / ( m β β Ψi i / β β whee Ψi β i / ( he β i / β (T.11 >0. Sice β / < 0 < 0. This implies ha he em i baces i (T.11 is less ha oe. Hece a sufficie codiio f / 0 < is ( / i 1 1 β m Λ>. To deive Poposiio s exac ad sufficie codiios f a geae umbe of Bs o educe he eail deposi aes of small bas i mae we ca diecly diffeeiae equaio (9 aleaively diffeeiae equaio (3 f he case of a small ba i mae : 3
i im / 1 1 = E c m 1 i To fid 1 / we diffeeiae 1 i equaio (8 o obai / m / / m / m / / 1 1 1 ( = m β / βm / β / βm / ( ( (T.1 β β β β β β Δ ( (T.13 Subsiuig i f 1 / ad 1 fom (T.13 ad equaio (8 (T.1 becomes im / 1 1 ( β ( i Δ = β β This deivaive is egaive whe β Δ ( β im / β β Δ ( β im / / β 1 1 im β β 0 > ( β β β β im / βm / β / Δ ( / βm / im / / m / / i (T.14 (T.15 im / β β β 1 1 β βm i ( β β Δ> im / β β β i which ca be wie as Δ> β β 1 1 Ψ im im / ( m β β Ψim (T.16 (T.17 4
im / β β whee Ψim β im / ( he β im / β >0. Sice β / < 0 < 0. This implies he em i baces i (T.17 is less ha oe. 1 1 Hece a sufficie codiio f / < 0 isδ> i ( β. / Similaly o fid he exac ad sufficie codiios f a geae umbe of Bs o educe he eail deposi aes of small bas i mae we ca diffeeiae equaio (3 f he case of a small ba i mae : i i / 1 1 = E c 1 i Subsiuig i f 1 / ad 1 fom (T.13 ad equaio (8 (T.18 becomes i / 1 1 ( β ( i Δ = β β This deivaive is egaive whe β Δ ( β i / m β β Δ ( β i / / β (T.18 (T.19 1 1 i / β β β 0 >( β β i βm / i / β β Δ ( β i (T.0 i / β β β 1 1 β βm i ( β β Δ> i / β β β i which ca also be wie as (T.1 5
Δ> β β 1 1 Ψ i i / ( m β β Ψi i / β β whee Ψi β i / ( he β i / β (T. >0. Sice β / < 0 < 0. This implies he em i baces i (T. (T.1 is less ha oe. Ideed f some values of ad his em i baces could eve become egaive i which case we would eed Δ > 0 f / < 0. This is because as meioed ealie (β β / - β β / > 0 f > m so ha he em i i / β β β β > 0 ad he umea of he em i baces i (T.1 ca become egaive f lage. Howeve fom ispecio of / i (T.19 oe ca see ha 1 1 β m he fis wo ems ae egaive whe ( / 1 1 β whe ( / i Δ> ad he hid em is egaive Δ>. Hece a sufficie codiio f / 0 < is ( / i 1 1 β Δ>. 6