3 The equilibrium rate of compliance
If each individual chooses the alternative with the highest expected utility, then compliance
is optimal for a type-(0, c) individual if and only if:
(1 - μp)U(R(A) - c) + μpU(V - f - c) > pU(V - F) + (1 - p)U(R(A)) - 0≠(λ),
or equivalently
■>-•» -+"
where
∆ll'(c, A) = pU(V - F) + (1 - p)U(R(A)) - (1 - μp)U(R(X) - c) - μpU(V - f - c)
is the expected utility gain from being non compliant as compared with compliance.
The following lemma indicates the basic properties of the minimal adherence to the norm
■ (c, A), needed for compliance to be optimal.
Lemma 1 The minimal adherence to the norm ■ (c, A) is continuous, decreasing in the com-
pliance rate A and increasing in the compliance cost c.
Proof: Indeed, we get
~
. O^
sign— = sign
OA
(° ∆∣∙ -ɪ(
Recall that rψ is increasing in A while ∆l is decreasing in A as shown below:
≤δ1 = [(1 - p)U'(R) - (1 - μp)U'(R - c)] 0r < 0
OA OA
because 1 - μp ≥ 1 - p and U'(R - c) > U'(R) as U is concave.
Moreover, we have sign∣∣ = sign {d^ } = sign {(1 - μp)U'(R(A) - c) + μpU'(V - f - c)} =
+ . This concludes the proof. ■
Intuitively, the minimal adherence to the norm for compliance to be chosen must be higher
when the compliance cost c increases. Furthermore, this threshold level decreases with the