on c (∙) imply that if C (0) < ψj (0) hhfhε Wj for j = 1, 2, then each first-order condition
has a unique and positive solution.15 Note also that if h1 = h2 and W1 = W2 , then
a*1 = a*2 .
Assuming from now onwards that an interior equilibrium indeed exists, we are inter-
ested in the comparative statics of the equilibrium effort levels with respect to the model
„ . , , da* ........... . .
parameters. Evidently, dwj > 0: the higher the prize of wining the nomination contest,
the higher the incentive to provide effort. With respect to the parameters characterizing
the prior belief distributions, we obtain the following results (the proof is relegated to the
appendix):
Proposition 1 (i) For every j = k ∈ {1, 2}, a* depends on ∆ = ∣m1
m1 and m2 individually, and
- m2 | but not on
da*
-< < 0 if ∆ > 0,
d∆
da*
—j = 0 if ∆ = 0.
d∆
(ii) Given (mj, mk, hk, hε), there exists for every j = k ∈ {1, 2} a
such that
7 7,7 . П
threshold hj > 0
daj*
—— = 0 if and only if hj = hj∙.
dhj
Similarly, given (mj, mk, hj, hε) there exists for every j = k ∈ {1, 2} a
such that
7 7 ,Г . Г
threshold hk > hj
daj > П . 7 ∙C 7 < T^
—0 = 0 if and only if hk = hk.
dhk
Moreover, limhι→0daj = ∞, and limhl→∞a* = 0.
j dhj j j
Figures 1 and 2 illustrate how a*1 varies with m1 and h1 , respectively, in an example.
The intuition for part (i) of the Proposition is straightforward. Ceteris paribus a smaller
difference ∆ between the mean prior reputations means that the race is closer in terms of
the expected winning probabilities being more equal. For each agent, a small increase in
effort is then more likely to affect the contest outcome and therefore more attractive.
15 The second-order equilibrium conditions are
ψj (0) f τhTΓ 1 Wj < c" (αj) for all j = k ∈ O’ 2} .
j hj + hε j
If mj = mk, then φj (0) = 0 for j = 1, 2, so the second-order conditions always hold in that case. In
what follows, we will assume that the second-order conditions are always satisfied.