residual MAI is instead impossible, because, when ° < °l, it occurs that ¢E <
0; when β = °: □
A.6 Appendix 6.
We prove here that there exist parameter constellations where countries gain
from participating to a partial MAI compared with what they could get in a
world without MAI. The condition for having a negative value for μPMZ is
p>
z
βh(1 - βh) ´ -
(βh - °)(1 - βh + °) ´ h:
(30)
We see that
@-h
@Ph
°[1 + ° - 2βh(1 - 2βh + °)] < 0:
[(βh - °)(1 - βh + °)]2
(31)
Recall that by (25) of appendix A.3
P =_______________βt (1 - β*)_______________
z (β* - °)(1 - β* + °)(1 - z) + zβ* (1 - β*) ;
wlicrc 0 — 0:5 β—
where z = 05-βl .
Note that lim p =lim —h =
^*!0:5 βh!!0∙5
---------1--------
4(0:5-°)(0:5+°) .
By evaluating the
total derivative of p with respect to β* = 0:5 we find
@ (P=z) 1 (1 - 2βι)(1 - 2βι - °)
(33)
_ _ ... --- --- “ :
@β '■' 0:5 2 (4 - 1 βι +2βι°2 - °2)
Since we are considering a partial MAI equilibrium, by Proposition 4 we need
°l <°<°u . The only ambiguous term in (33) is (1 - 2βl - °), which is
decreasing in °. Plugging the highest possible value for ° (i.e., °u) into this
term we find
1 - 2βι- -~ 2+6 q21+12β2 - 3oβι; (34)
which is
always positive for βl < 0:5. It follows that @@p=*z)
j^*=0:5
< 0. So, one
can always choose a pair (°, βι) such that β* < 1=2 and some countries with βh
sufficiently close to 1/2 for which p=z > —h. □
25
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