Effort and Performance in Public-Policy Contests

own-stake (“income”) effect, when i=j. The former effect is ambiguous, depending
on the sign of the cross-partial derivative of the contest success function. The latter
effect is clear-cut, due to our assumption that the marginal winning probability of a
contestant is declining in his own effort.

In case (ii) with i=H and case (iii) with i=L,

In cases (iii) with i=H and case (ii) with i=L,

We therefore obtain

Proposition 1:

In case (ii) with i=H and case (iii) with i=L,

∂ x ^*

Sig^π(      )Sig^π (ηh ) and Sign(

∂I

In cases (iii) with i=H and case (ii) with i=L,

*     *2

^Sign⁽      )Sign (ηL ) and Sign(--^)=Sign ⁽     ” ηL ).

∂ I                      ∂ I         ∂x ∂x

HL

Proposition 1 directly yields the following general comparative statics result that
focuses on the sensitivity of a contestant’s effort to a change in his or his rival’s stake:

∂n_H          ∂x_L ∂x_H

By this first corollary, under our general contest success function, the effort exerted
by a contestant is positively related to his stake. That is, the strategic own-stake
(“income”) effect is always positive (effort of every player is a “normal good”). In
contrast, the effort exerted by a player can be positively or negatively related to the
stake of his rival. It can also be independent of the rival’s stake. When the marginal

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