Dynamic Certification and Reputation for Quality Online Appendix

Proof. Let pa ≡ sup{p ∈ [0, 1] : a(p) > 0}, pc ≡ sup{p ∈ [0, 1] : d(p,H) = 1}, τa ≡ inf{t > 0 : pt = pa, p0 = 1}, and τc ≡ inf{t > 0 : pt = pc, p0 = 1}. First, we show that in any equilibrium pa ≤ pc. Looking for a contradiction, suppose that pa > pc. Let’s consider the behavior of beliefs at the threshold pa. If a(pa) ≥ pa then λ(a(pa) − pa) ≥ 0 so beliefs never cross the threshold pa. On the other hand, if a(pa) < pa then beliefs cross the threshold pa however if this is the case, we have that k/λ = D(pa) = e caD(pc) < e cD(pc) = D(pt) for all t ∈ (τa, τc]. This means that a(pa− ) = 1 but if this is the case then beliefs can never cross the threshold pa. This in turn implies that τc = ∞, so that D(pt) = ecD(pc) = 0. This contradicts the hypothesis that pa > pc ≥ 0 which requires that λD(pa) ≥ k. Second, we analyze the certification strategy. By definition we have that d(p, θ) = 0 for