3.6 Extensions
3.6.4 Partially Asymmetric Information
In this modification of the main text version of the private vetting model, the politician observes σ, which contains both the public knowledge in the vetting agency’s report about her policy and her own knowledge of her competence. However, the voter only observes the public vetting about the politician, which indicates the expected probability that she is competent, πτ. Whenever πτ > π0τ, I assume that the distribution of σ givenπτ strictly first order stochastically dominates (FOSDs) the distribution givenπτ0 on[0,1]. I also add the assumption thatσpri is unique, as the proof of uniqueness used in the main text requires that the expectedσ be equal to 12. All of the other assumptions are maintained from Section 3.3.
Asπτ increases, the voter expects the politician to have better signals, and consequently the expected signal for any given cutoff is higher.
Lemma 3.8. If πτ > π0τ,ER(σ∗|πτ)> ER(σ∗|πτ0).
Proof. Let the distribution for σ given πτ beg and the distribution givenπ0τ be h. By assumption, g strictly FOSDsh on [0,1]. Using the definition ofER, it must be shown that
Rσ∗
0 σdG(σ) +γR1
σ∗σdG(σ)
>
Rσ∗
0 σdH(σ) +γR1
σ∗σdH(σ)
. (3.93)
First order stochastic dominance implies that Z σ∗
0
σdG(σ)>
Z σ∗ 0
σdH(σ) (3.94)
and
Z 1 σ∗
σdG(σ)>
Z 1 σ∗
σdH(σ). (3.95)
Consequently, the numerator of the left hand side of (3.93) must be larger than the numerator of the right hand side. Further, G(σ∗) < H(σ∗) for any σ∗ by first order stochastic dominance. Hence, the denominator of the left hand side of (3.93) must be smaller than that of the right hand side as well.
Recall that σpri solves F−1(σpri) = ER(σpri). Hence, when ER(σpri) increases because of an increase in πτ, F−1(σpri) must also increase. Consequently, because F−1(σ) is strictly increasing, the value of σpri must be greater for πτ than forπ0τ. Hence, as πτ increases, the value of σpri increases. When the politician has a higherπτ, she implements the policy only for larger realizations of σ. Because σ is the probability of success, the politician is effectively becoming more risk averse asπτ increases, as a strictly higher probability is required for the lottery of policy implementation to be acceptable to her.
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