Appendix D: Proofs of Proposition A1 and Claim 1

Appendix D: Proofs of Proposition A1 and Claim 1

Proof of Proposition A1. (i) From (43) in the proof of Lemma 2, d[wn(e⁺, sl)hn(e, sl)]

ds_l =wn(e⁺, s_l)∂hn(e, sl)

∂s_l +dwn(e⁺, sl)

ds_l hn(e, s_l) =hn(e, s_l)

∙

−wn(e⁺, sl)

1−s_l +dwn(e⁺, sl) ds_l

¸

=w_n(e⁺, s_l)h_n(e, s_l) 1−s_l

⎧⎨

⎩−1+(1−α)e⁺ h_l(e⁺, s_l)

α_e^h+^l

h

h_lF(e⁺)+δ_lRe⁺

0 ef(e)dei +³_h

l e⁺+δ_l´

h_l(e⁺, s_l)e⁺f(e⁺) (1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]ei

+h_l(e⁺, s_l)e⁺f(e⁺)

⎫⎬

⎭

=w_n(e⁺, s_l)h_n(e, s_l) 1−s_l

1 h_l(e⁺, s_l)

⎛

⎝ −hl(e⁺, sl) n

(1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]e i

+hl(e⁺, sl)e⁺f(e⁺) o

+(1−α)e⁺n α_e^h+^l

h

h_lF(e⁺)+δ_lRe⁺

0 ef(e)dei +³_h

l e⁺+δ_l´

h_l(e⁺, s_l)e⁺f(e⁺)o

⎞

⎠ (1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]ei

+h_l(e⁺, s_l)e⁺f(e⁺)

=wn(e⁺, s_l)hn(e, s_l) 1−s_l

1 h_l(e⁺, s_l)

Ã(1−α)_e^h+^l

n αe⁺h

h_lF(e⁺)+δ_lRe⁺

0 ef(e)dei

−h_l(e⁺, s_l)hRe

e⁺ef(e)de+[1−F(e)]eio +h_l(e⁺, s_l)e⁺f(e⁺) [(1−α)(h_l+δ_le⁺)−h_l(e⁺, s_l)]

!

(1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]ei

+h_l(e⁺, s_l)e⁺f(e⁺)

=wn(e⁺, sl)hn(e, sl) 1−s_l

1 h_l(e⁺, s_l)

Ã

(1−α)h_lαnh

h_lF(e⁺)+δ_lRe⁺

0 ef(e)dei

−₁₋¹_αh

h_lF(e⁺)+δ_ls_lRe⁺

0 ef(e)deio +h_l(e⁺, s_l)e⁺f(e⁺) [(1−α)(h_l+δ_le⁺)−h_l(e⁺, s_l)]

!

(1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]e i

+hl(e⁺, sl)e⁺f(e⁺)

³

because h_l(e⁺, s_l)hRe

e⁺ef(e)de+[1−F(e)]ei

=₁₋^α_αh δ_ls_lRe⁺

0 ef(e)de+h_lF(e⁺)i

e⁺from (7)´

=wn(e⁺, s_l)hn(e, s_l) 1−s_l

1 h_l(e⁺, s_l)

à αh_l

n (1−α)

h

h_lF(e⁺)+δl

Re⁺

0 ef(e)de i

− h

h_lF(e⁺)+δlsl

Re⁺

0 ef(e)de io +h_l(e⁺, s_l)e⁺f(e⁺) [−αh_l+(1−α−s_l)δ_le⁺]

!

(1−α)_e^h+^l

hRe

e⁺ef(e)de+ [1−F(e)]ei

+h_l(e⁺, s_l)e⁺f(e⁺) .

(D1) [Proof for ^d(w_ds^nhn)

l <0] Whens_l≥1−α⇔1−α−s_l≤0, ^d(w_ds^nhn)

l <0 because the expressions inside the big parenthesis of (D1) is negative.

When s_l < 1−α, ^d(w_ds^nhn)

l < 0 if −αh_l+(1−α−s_l)δ_le⁺ ≤0 because −αh_l+(1−α−s_l)δ_le⁺ >

1 F(e⁺)

n (1−α)h

h_lF(e⁺)+δ_lRe⁺

0 ef(e)dei

−h

h_lF(e⁺)+δ_ls_lRe⁺

0 ef(e)deio

for the expressions inside the paren- thesis. −αh_l+(1−α−s_l)δ_le⁺ ≤0 holds iff s_l ≥(1−α)−α_δ ^hl

le⁺(sl), wheree⁺(s_l) is, from (7) and (5), a solution for

α 1−α

h

h_lF(e⁺)+δ_ls_lRe⁺

0 ef(e)de i

e⁺=hRe

e⁺ef(e)de+ [1−F(e)]e i

h_l(e⁺, s_l) (D2)

⇔δ_ls_le⁺hRe

0ef(e)de+[1−F(e)]e−₁₋¹_αRe⁺

0 ef(e)dei

=n

α

1−αF(e⁺)e⁺+Re⁺

0 ef(e)de−hRe

0ef(e)de+[1−F(e)]eio h_l. (D3) s_l−h

(1−α)−α_δ ^hl

le⁺(sl)

i

increases with s_l because 1−α h_l

δ_l(e⁺)² de⁺

ds_l

>1−α h_l δ_l(e⁺)²

δ_l(e⁺)²

h_l >0.(from (37) in the proof of Lemma 1). (D4)

Further, at s_l =s^??_l ≡ (1−α)−α_δ^hl

le, s_l >(1−α)−α_δ ^hl

le⁺(s_l). Hence, if (1−α)−α_δ ^hl

le⁺(0) ≤ 0⇔e⁺(0)≤ ₍₁₋^αh_α)δ^l _l,(1−α)(h_l+δ_le⁺)−h_l(e⁺, s_l)≤0 holds for anys_l and thus ^d(w_ds^nhn)

l <0 for any sl. e⁺(0) is the solution to ₁₋^α_αF(e⁺)e⁺+Re⁺

0 ef(e)de=Re

0ef(e)de+[1−F(e)]efrom (D3), thus this is the case when many individuals have limited wealth.

Otherwise, i.e. e⁺(0)> ₍₁₋^αh_α)δ^l

l,there exists uniques^]_l ∈(0, s^??_l ) satisfyings^]_l = (1−α)−α ^hl

δle⁺(s^]_l)

and−αh_l+(1−α−s_l)δ_le⁺≤0 fors_l ≥s^]_l.Thus, ^d(w_ds^nhn)

l <0 fors_l≥s^]_l.(Note that this is a sufficient but not necessary condition. ^d(w_ds^nhn)

l <0 could hold for smaller s_l or for anys_l,if the expressions inside the big parenthesis of (D1) is negative.)

[Proof for ^d(w_ds^nhn)

l >0] As shown above, ^d(w_ds^nhn)

l >0 is possible only whens_l<1−α,in which case −αh_l+ (1−α−s_l)δ_le⁺ > _F(e¹+)

n (1−α)h

h_lF(e⁺)+δ_lRe⁺

0 ef(e)dei

−h

h_lF(e⁺)+δ_ls_lRe⁺

0 ef(e)deio holds. Thus, ^d(w_ds^nhn)

l > 0 when (1−α)h

h_lF(e⁺)+δ_lRe⁺

0 ef(e)dei

−h

h_lF(e⁺)+δ_ls_lRe⁺

0 ef(e)dei

≥ 0, which holds iffs_l≤1−α−α ^hlF(e+(sl))

δl

Re⁺(sl)

0 ef(e)de

,where e⁺(s_l) is a solution to (D3).

Ats_l=s^??_l ≡(1−α)−α_δ^hl

le, s_l>(1−α)−α ^hlF(e+(sl))

δ_lRe⁺(sl)

0 ef(e)de

.Further,s_l−

"

(1−α)−α^h_δ^l

l

F(e⁺(sl))

Re⁺(sl)

0 ef(e)de

#

increases with sl if Re⁺(0)

0 ef(e)de

F(e⁺(0)) ≥ 1+α^α e⁺(0).

This can be proved as follows. The derivative of the expression with respect to sl equals 1−αh_l

δ_l

F(e⁺)e⁺−Re⁺ 0 ef(e)de (Re⁺

0 ef(e)de)²

f(e⁺)de⁺ ds_l

= 1−αh_l δ_l

F(e⁺)e⁺−Re⁺

0 ef(e)de hRe⁺

0 ef(e)dei2 f(e⁺) δ_le⁺n

(1−α)hRe

e⁺ef(e)de+[1−F(e)]ei

−αRe⁺

0 ef(e)deo (1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]ei

+h_l(e⁺, s_l)e⁺f(e⁺)

(from (37))

=

⎛

⎜⎝

½

1 e⁺

hRe⁺

0 ef(e)dei2

−αh

F(e⁺)e⁺−Re⁺

0 ef(e)dei

f(e⁺)e⁺

¾

(1−α)h_lhRe

e⁺ef(e)de+[1−F(e)]ei +hRe⁺

0 ef(e)de i2

hl(e⁺, sl)e⁺f(e⁺)+α²h_l h

F(e⁺)e⁺−Re⁺

0 ef(e)de i

f(e⁺)e⁺hRe⁺

0 ef(e)de i

⎞

⎟⎠

hRe⁺

0 ef(e)de i2n

(1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]e i

+hl(e⁺, sl)e⁺f(e⁺) o

=

⎛

⎜⎝

½

1 e⁺

hRe⁺

0 ef(e)dei2

−αh

F(e⁺)e⁺−Re⁺

0 ef(e)dei

f(e⁺)e⁺

¾ αh_lh

h_lF(e⁺)+δ_ls_lRe⁺

0 ef(e)dei

e⁺ hl(e⁺,sl)

+hRe⁺

0 ef(e)dei2

h_l(e⁺, s_l)e⁺f(e⁺)+α²h_lh

F(e⁺)e⁺−Re⁺

0 ef(e)dei

f(e⁺)e⁺hRe⁺

0 ef(e)dei

⎞

⎟⎠

hRe⁺

0 ef(e)dei2n

(1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]ei

+h_l(e⁺, s_l)e⁺f(e⁺)o (from (D2))

=

⎛

⎜⎜

⎝

½

1 e⁺

hRe⁺ 0 ef(e)de

i2

−α h

1−hl(e⁺, sl)_δ ¹

lsle⁺

ih

F(e⁺)e⁺−Re⁺

0 ef(e)de i

f(e⁺)e⁺

¾ αh_l

³ δlsl

Re⁺

0 ef(e)de

´ e⁺ hl(e⁺,sl)

+

½

1 e⁺

hRe⁺

0 ef(e)dei2

−αh

F(e⁺)e⁺−Re⁺

0 ef(e)dei f(e⁺)e⁺

¾

αh_l[h_lF(e⁺)]_h ^e+

l(e⁺,sl)+hRe⁺

0 ef(e)dei2

h_l(e⁺, s_l)e⁺f(e⁺)

⎞

⎟⎟

⎠ hRe⁺

0 ef(e)dei2n

(1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]ei

+h_l(e⁺, s_l)e⁺f(e⁺)o

=

⎛

⎝

hRe⁺

0 ef(e)dei2h δ_ls_lRe⁺

0 ef(e)de+h_lF(e⁺)i αh_l +

n

[hl(e⁺, sl)−αh_l]hRe⁺

0 ef(e)de i

+αh_lF(e⁺)e⁺ on

[hl(e⁺, sl)+αh_l]hRe⁺

0 ef(e)de i

−αh_lF(e⁺)e⁺ o

f(e⁺)e⁺

⎞

⎠ hl(e⁺, sl)hRe⁺

0 ef(e)de i2n

(1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]e i

+hl(e⁺, sl)e⁺f(e⁺)

o .

(D5) The expression is positive if

£hl(e⁺, sl)+αh_l¤hRe⁺

0 ef(e)de i

−αh_lF(e⁺)e⁺≥0

⇔n 2hRe⁺

0 ef(e)dei

−(1−α)hRe

0ef(e)de+[1−F(e)]eio

h_l(e⁺, s_l)≥0, (D6) where the second equality is from (D2), which can be expressed as αh_lh

F(e⁺)e⁺−Re⁺

0 ef(e)dei + h_l(e⁺, s_l)Re⁺

0 ef(e)de= (1−α)hRe

0ef(e)de+[1−F(e)]ei

h_l(e⁺, s_l).The inequality holds for anys_liff 2hRe⁺(0)

0 ef(e)dei

−(1−α)hRe

0ef(e)de+[1−F(e)]ei

≥0

⇔ Re⁺(0)

0 ef(e)de

F(e⁺(0)) ≥ _1+α^α e⁺(0) (from (D3)). (D7) Hence, if (1−α)−α ^hlF(e+(0))

δ_lRe⁺(0) 0 ef(e)de

>0⇔ Re⁺(0)

0 ef(e)de

F(e⁺(0)) > ₍₁₋^αh_α)δ^l

l and

Re⁺(0)

0 ef(e)de

F(e⁺(0)) ≥ _1+α^α e⁺(0), there exists uniques^[_l ∈(0, s^]_l) satisfyings^[_l = (1−α)−α ^hlF(e

+(s^[_l)) δ_lRe⁺(s^[_l)

0 ef(e)de

and ^d(w_ds^nhn)

l >0 fors_l≤s^[_l. (ii) From (43) in the proof of Lemma 2,

d[wl(e⁺, sl)hl(e, sl)]

ds_l =w_l(e⁺, s_l)∂hl(e, sl)

∂s_l +dwl(e⁺, sl)

ds_l h_l(e, s_l)

=w_l(e⁺, s_l)δ_le− α 1−α

w_l(e⁺, s_l) w_n(e⁺, s_l)

dwn(e⁺, s_l)

ds_l h_l(e, s_l)

=w_l(e⁺, s_l)

⎧⎨

⎩δ_le−α h_l(e, s_l) h_l(e⁺, s_l)

e⁺ 1−s_l

α_e^h+^l

h

h_lF(e⁺)+δ_lRe⁺

0 ef(e)dei +³_h

l e⁺+δ_l´

h_l(e⁺, s_l)e⁺f(e⁺) (1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]ei

+h_l(e⁺, s_l)e⁺f(e⁺)

⎫⎬

⎭

=w_l(e⁺, s_l)

⎛

⎝δ_leh_l(e⁺, s_l)(1−s_l)n

(1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]ei

+h_l(e⁺, s_l)e⁺f(e⁺)o

−αh_l(e, s_l)n αh_lh

h_lF(e⁺)+δ_lRe⁺

0 ef(e)dei

+ (h_l+δ_le⁺)h_l(e⁺, s_l)e⁺f(e⁺)o

⎞

⎠ h_l(e⁺, s_l)(1−s_l)n

(1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]ei

+h_l(e⁺, s_l)e⁺f(e⁺)o

=w_l(e⁺, s_l)

⎧⎨

⎩

δ_leh_l(e⁺, s_l)(1−s_l)(1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]ei

−αh_l(e, s_l)αh_lh

h_lF(e⁺)+δ_lRe⁺

0 ef(e)dei

+h_l(e⁺, s_l)e⁺f(e⁺)[δ_leh_l(e⁺, s_l)(1−s_l)−αh_l(e, s_l)(h_l+δ_le⁺)]

⎫⎬

⎭ h_l(e⁺, s_l)(1−s_l)n

(1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]ei

+h_l(e⁺, s_l)e⁺f(e⁺)o

=wl(e⁺, sl) Ã

αh_ln

δ_le(1−s_l)h

h_lF(e⁺)+δ_ls_lRe⁺

0 ef(e)dei

−αh_l(e, s_l)h

h_lF(e⁺)+δ_lRe⁺

0 ef(e)deio +h_l(e⁺, s_l)e⁺f(e⁺) [δ_le(1−s_l)h_l(e⁺, s_l)−αh_l(e, s_l)(h_l+δ_le⁺)]

!

hl(e⁺, sl)(1−sl) n

(1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]e i

+hl(e⁺, sl)e⁺f(e⁺)

o (from (D2))

=wl(e⁺, sl) Ã

αh_ln

[(1−s_l−αs_l)δ_le−αh_l]h_lF(e⁺)+[(1−s_l−α)δ_ls_le−αh_l]δ_lRe⁺

0 ef(e)deo +h_l(e⁺, s_l)e⁺f(e⁺){[(1−s_l−αs_l)δ_le−αh_l]h_l+[(1−s_l−α)δ_ls_le−αh_l]δ_le⁺}

!

hl(e⁺, sl)(1−sl) n

(1−α)_e^h+^l

hRe

e⁺ef(e)de+[1−F(e)]e i

+hl(e⁺, sl)e⁺f(e⁺)

o . (D8)

[Proof for ^d(w_ds^lhl)

l >0] If the expression inside the big parenthesis of (D8) is positive, (1−s_l− αs_l)δ_le−αh_l>0 and thus [(1−s_l−αs_l)δ_le−αh_l]Re^h+lF(e+)

0 ef(e)de

+[(1−s_l−α)δ_ls_le−αh_l]δ_l>[(1−s_l−αs_l)δ_le−αh_l]_e^h+^l+ [(1−s_l−α)δ_ls_le−αh_l]δ_lhold. Hence, ^d(w_ds^lhl)

l >0 if [(1−s_l−αs_l)δ_le−αh_l]_e^h+^l+[(1−s_l−α)δ_ls_le−αh_l]δ_l≥ 0⇔L(s_l)≡−ee⁺(s_l)s²_l+h

(1−α)e⁺(s_l)−(1+α)^h_δ^l

l

i es_l+h

−α³_h

l

δl+e⁺(s_l)´ +ei_h

l

δl≥0⇔s_l∈h

max(0, s⁴_l,l(e)), s⁴_l,h(e)i , wheres⁴_l,h(e) (s⁴_l,l(e)) is the solution to

s_l= h

(1−α)e⁺(s_l)−(1+α)^h_δ^l

l

i

e+ (−) rh

(1−α)e⁺(s_l)−(1+α)^h_δ^l

l

i2

e²+4ee⁺(s_l) h

−α

³_h

l

δl+e⁺(s_l)

´ +e

i_h

l δl

2ee⁺(sl)

= h

(1−α)e⁺(s_l)−(1+α)^h_δ^l

l

i

+ (−)rh

(1−α)e⁺(s_l)−(1+α)^h_δ^l

l

i2

+4e⁺(s_l)h

−α³_h

l

δl+e⁺(s_l)´

e⁻¹+1i_h

l δl

2e⁺(s_l)

= h

(1−α)−(1+α)_δ ^hl

le⁺(s_l)

i + (−)

vu uth

(1−α)−(1+α)_δ ^hl

le⁺(s_l)

i2

+4

"

−α+^e−α

hl δl e⁺(s_l)

#

h_l δle

2 . (D9)

Real solutions to (D9) could exist only if the expression inside the square root is non-negative (from the second equation),

e≥Λ(e⁺(s_l))≡

4α³_h

l

δl+e⁺(s_l)´_h

l δl

4^h_δ^l

l+_e+¹(sl)

h

(1−α)e⁺(s_l)−(1+α)^h_δ^l

l

i2 =

α³_h

l

δl+e⁺(s_l)´ 1+_4h ^δl

le⁺(sl)

h

(1−α)e⁺(s_l)−(1+α)^h_δ^l

l

i2, (D10) which holds for any s_l if

e≥Λ(e)≡

α³_h

l δl+e´ 1+_4h^δl

le

h

(1−α)e−(1+α)^h_δ^l

l

i2, (D11)

because the derivative of (D10), which can be expressed as

4α³ 1+_δ ^hl

le⁺(sl)

´_h

l δl

4_δ ^hl

le⁺(sl)+h

(1−α)−(1+α)_δ ^hl

le⁺(sl)

i2, with respect tos_l is proportional to

−

½ 4h_l δle⁺(sl)+

h

(1−α)−(1+α)_δ ^hl

le⁺(sl)

i2¾

h_l

δl[e⁺(sl)]²e⁺⁰(s_l)−

³ 1+_δ ^hl

le⁺(sl)

´nh

(1−α)−(1+α)_δ ^hl

le⁺(sl)

i

(1+α)−2 o _2h

l

δl[e⁺(sl)]²e⁺⁰(s_l)

=−

½

(1−α)²+h

(1+α)_δ ^hl

le⁺(sl)

i2

+2(1+α²)_δ ^hl

le⁺(sl)−2³ 1+_δ ^hl

le⁺(sl)

´h

(1+α)²_δ ^hl

le⁺(sl)+(1+α²)i¾ _h

l

δl[e⁺(sl)]²e⁺⁰(s_l)

=−

½

(1−α)²−h

(1+α)_δ ^hl

le⁺(s_l)

i2

−2(1+α²)−2(1+α)²_δ ^hl

le⁺(s_l)

¾

h_l

δ_l[e⁺(s_l)]²e⁺⁰(s_l)

= (1+α)²

³ 1+_δ ^hl

le⁺(sl)

´2 h_l

δl[e⁺(sl)]²e⁺⁰(sl)>0. (D12)

s⁴_l,h(e)∈(0,1) satisfying (D9) exists and is unique whene >α³_h

l

δl+e⁺(0)´

ande≥Λ(e) or when e⁺(0)>^1+α_1−α^h_δ^l

l and e≥Λ(e), because [1] the RHS of (D9) decreases withe⁺(s_l) and thuss_l,[2] the RHS at s_l = 0 is greater than 0,which holds clearly for the above ranges ofe from (D9), and [3]

the RHS ats_l= 1 is smaller than 1.

[1] can be proved as follows. The derivative of the numerator of the RHS of (D9) with respect toe⁺ equals

(1+α)h_l δl[e⁺(sl)]²+(−)

(h

(1−α)−^(1+α)h_δle⁺_(sl)^l i2

+4 Ã

−α+^e−α

hl δl e⁺(sl)

!

h_l δ_le

)₋¹₂(h

(1−α)−^(1+α)h_δle⁺_(sl)^l

i _(1+α)h

l

δl[e⁺(sl)]² −2 ^e−α

hl δl [e⁺(sl)]²

h_l δ_le

)

= (h

(1−α)−^(1+α)h_δle⁺_(sl)^li2

+4 Ã

−α+^e−α

hl δl e⁺(sl)

!

h_l δle

)₋¹₂

h_l δ_l[e⁺(s_l)]²

⎛

⎜⎜

⎜⎜

⎝ (h

(1−α)−^(1+α)h_δle⁺_(sl)^l i2

+4 Ã

−α+^e−α

hl δl e⁺(sl)

!

h_l δle

)¹

2

(1+α) +(−)

(h

(1−α)−^(1+α)h_δle⁺_(sl)^l i

(1+α)−2^e−α

hl δl e

)

⎞

⎟⎟

⎟⎟

⎠.

(D13) The second term inside the big parenthesis of the above equation is negative because

h

(1−α)−^(1+α)h_δle⁺_(sl)^li

(1+α)−2^e−α

hl δl e

≤h

(1−α)− ^(1+α)h_δle⁺_(sl)^li

(1+α)−2+2α^h_δ^l

l

⎧⎪

⎨

⎪⎩

4α³ 1+_δ^hl

le

´_h

l δl

h

(1−α)−(1+α)_δ^hl

le

i2

+4_δ^hl

le

⎫⎪

⎬

⎪⎭

−1

(from (D11))

<−h_(1+α)2h_l

δle +(1+α²)i 2³

1+_δ^hl

le

´ +h

(1−α)−(1+α)_δ^hl

le

i2

+4_δ^hl

le

2³ 1+_δ^hl

le

´

=− h

(1+α)_δ^hl

le

i2

−2^(1+α)_δ ^2hl

le −(1+α²)2

³ 1+_δ^hl

le

´ +

h

(1−α)²−2(1−α²)_δ^hl

le

i +4_δ^hl

le

2³ 1+_δ^hl

le

´

=−^(1+α)₂ ²³ 1+_δ^hl

le

´

<0. (D14)

Then, the RHS of (D9) decreases with e⁺ when the solution is s⁴_l,h(e) because (h

(1−α)−^(1+α)h_δle⁺_(sl)^l i2

+4 Ã

−α+^e−α

hl δl e⁺(sl)

!

h_l δ_le

)

(1+α)²<

(

− h

(1−α)−^(1+α)h_δle⁺_(sl)^l i

(1+α)+2^e−α

hl δl e

)2

⇔ Ã

−α+^e−α

hl δl e⁺(sl)

!

h_l

δ_le(1+α)² <^e−α

hl δl e

(

− h

(1−α)−(1+α)_δ ^hl

le⁺(sl)

i

(1+α)+^e−α

hl δl e

)

⇔ −^h_δl^l(1+α)²<(e−α^h_δ^l

l)(α−_δ^h_l^l_e)

⇔ −2h_l

δ_l < e+ h_l δ_l

h_l

δ_le. (D15)

[3] is true since