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

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.


12Prj

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
η
i nj ïï

J


ij, 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
ij, or to the strategic

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