A Appendix
A.1 Appendix 1. Proof of Lemma 1.
From the definition of ? we have
μ ⅛ ) ? (ɪ - ?)=( p )(? - °χι - ?+°). (15)
Next, from the partial derivatives of expected countries’ income with respect to
?h at given z and p
@EZ yh
@?h
@E-Z yh
@?h
0.5 (zp) (ɪ - 2?h + 2°) ; (16)
0∙5 (1≡7) (ɪ - 2?h) ; (17)
we can establish that @E--Z yh < @EZ yh if and only if
@Ph @fih j
(? - °) (ɪ - ? + °) (ɪ - 2?h) < ? (ɪ - ?) (ɪ - 2?h + 2°) ; (18)
where (15) has been used in (16) and (17). Inequality (18) is clearly satisfied,
since ?<0.5 and ?h < 0.5 for all h. The result of Lemma 1 follows directly
from (15) and (18). □
A.2 Appendix 2: Proof of Lemma 2.
Using (8) and developing integrals the difference ¢E = EZ¼ - EZ¼ can be
evaluated as follows:
-5 + 18° + 12°2 + ?(2 - 12° - 4?l) + 12?i - 4?2
¢^ =------------------------------------
; (19)
24
where ?l ∙ ? ∙ 0.5. It is easily checked that the value ?* that equates to zero
¢E is given by
15 - 18° - 12°2 - 12?l + 4?2
(20)
2 ɪ - 6° - 2?l
One can see from (20) that ?* as a function of ° has an asymptote in ° =
0:5-f9l ´ °*. Moreover, ɪ- = 61 3'9l+2'9l+6° +4fl° 2° > 0 in the admissible
3 ° , @° (1-6°-2βl)2 —
range of ?l and °, so that lim ?* = +ι, and lim ?* = -1. Since, when ° = 0,
°!°*- °!°* +
?* is always higher than 1=2, we necessarily have that ?* > 1=2 for ° < °*.
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