For Whom is MAI? A theoretical Perspective on Multilateral Agreements on Investments

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

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
¢^ =------------------------------------

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

2 ɪ - 6° - 2?l

One can see from (20) that ?* as a function of ° has an asymptote in ° =
0:5-^f9l ´ °*. Moreover, ɪ- = 6^{1 3}'⁹l⁺2'⁹l⁺⁶° ⁺4f^l° ²° > 0 in the admissible
3        °                ^, @°               (1-6°-2β_l)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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