Effort and Performance in Public-Policy Contests



(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

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