(7)
∂ x * i
~iΓ^
ni
i2 Pri iPrj inj i2 Prj iPri ini
ixiixj ixj iI j ixj2 ixi iI
( ∂2 Prj ∂2 Pr. ∂2 Pr. ∂2 Prj 'ï
i- i
∂x 2 ∂x.2 ∂x i ∂x ∂x i ∂x
j . .j .j
∀.≠ j, .,j=L,H
Rewriting (7) together with (5), we obtain the fundamental equation that generates all
the comparative statics results:
(8)
∂ x*.
∂I
ɪ ∂2 Pri
B ∂xi ∂x
.j
ηjn.
1∂2Prj
B ∂xj 2
η. nj
∀.≠ j, .,j=L,H
where B =I n. n
i Pr
I dx j
∂2 Pr.
∂x.2
i Pr i Pr Ï
∂x i ∂x. ∂x i ∂x.
i j i j J
and all second-order partial
derivatives are computed at the Nash equilibrium (x *H , x*L) . The first term in (8)
represents the strategic rival’s-stake (“substitution”) effect. The sign of this term is
∂2 Pr.
equal to the sign of------η j. The second term represents the own-stake ( income )
∂x. ∂x
.j
effect. The sign of this term is equal to the sign of η . . By assumption,
∂2 Pr. (x. , x )
----i±÷j < 0 and, by (2),
∂ x. 2
i 2 Pr-(xi, xj )
∂ x. ∂ x j
12Prj(xj, xi)
∂ x. ∂ x j
< 0 . Hence, B>0.
A. . Public Policy, Efforts and Winning Probabilities
When a change inI only affects the stake of one of the contestants, as in reforms type
(ii) and (iii), η . or η j is equal to zero and (8) reduces to
(8’)
∂x*.
∂I
( ∂ 2Pri ï
--------η j ni
∂x i ∂x,
ij
or
∂x*i
∂I
∂2 Prj
∂xj2
ï
ηi nj ïï
J
∀i≠ j, i,j=L,H
In these cases the change in player i’s effort corresponding to the change in nj is
equal to the strategic rival’s-stake (“substitution”) effect, when i≠j, or to the strategic
11
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