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

Furthermore, one easily checks that when ° = 1/2, then β* = 11 ⁷⁺¹₂+¾_fg^4β is
always larger than 7/4. Hence, there must exist °_l and °_u, 0 < °* < °_l < °_u <
1/2, such that β_l < β* < 0.5 if and only if °_l < ° < °_u. The values of °_l and
°_u are found to be, respectively,

for β < 0.5. Consider then β = 0.5. In this case, ΔE = 0 iff. ° = °_u. Thus, by
(23), it must be that ΔE > 0 if and only if ° < °_u. Likewise, consider β = β_l.
Then, ΔE = 0 iff. ° = °_l. Thus, by (23), it must be that ΔE < 0 if and only
if ° > °_l. The above findings are summarized in Lemma 2. □

A.3 Appendix 3. Proof of Lemma 3.

Since EZyh = E_ Zy_h for the country with bargaining power β*, we have that

(^p´ (β* - °)(1 - β* + °) - ( — ɔ (β* (1 - β*)) = 0.       (24)

∖Z∕                              ∖ 1 — z J

Solving (24) for p we obtain

zβ* (1 - β*)

(β* - °)(1 - β* + °)(1 - z) + zβ* (1 - β*).

It is directly seen from (25) that p is in the range (0,1) for any z in (0,1) and
that limp = 0 and limp = 1. It can also be checked that since β* < 1/2, then
z→0           z→1

p > z. Finally, we can show that p is monotonically rising with z. To do that,
we need to recall that β* = 0.5 - z (0.5 - β_l) (this comes from (7)). >From
total differentiation of p with respect to z we have @p = p_z + pj- β). Partial
derivatives are easily evaluated as follows:

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