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,
*
∂xH 1(
∂2 PrL
∂I B
I ∂Xl 2
Ï
η h nL
J
and
∂ x * L
∂I
BI ∂x j ∂x„
LH
ηn
HL
In cases (iii) with i=H and case (ii) with i=L,
∂I
P-
√'x H dx L
ηn
LH
and
∂I
∂2 PrH
2HηL
∂xH
λ∣
nH
J
We therefore obtain
Proposition 1:
In case (ii) with i=H and case (iii) with i=L,
∂ x *
∂∣
L ∂2 Pr
l- )=Sign ( —--L-ηH ).
∂x ∂x
LH
∂ x *
Sigπ( )Sigπ (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:
|
∂ H 1 1 ∂ x * H Corollary 1.1: ----- dnH |
> 0, |
∂ x * L |
> 0, |
∂ x *H Sign (—-- ∂nL |
∂2 P- ) = Sign ( —ʒɪ ) ∂x ∂x HL |
|
and |
Sign( |
∂ x * L ) |
∂2 Pr = Sign ( —-L- ). | ||
∂nH ∂xL ∂xH
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
12
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