The name is absent

from insurers are inimical to the interests of policyholders. While, of course, there are other dimensions
to the contractual relationships not addressed here, and compensation also must address these, our model
shows that contingent fees can lead to an expansion of insurance markets to include informal coverage of
non verifiable losses and this is beneficial to policyholders.¹³

AAppendix:Proofs

A.1 Proof of Proposition 1

The policyholder solves the following program

max E [u (w)] = (1 - p) u (w0 - P)+pqu (w0 - P - L + c)+p (1 - q) u (w0 - P - L + b)

c,b

subject to

P = pqc + p (1 - q) b + rb.

The first derivative of expected utility with respect to b is

- (1 - p)(p (1 - q)+r) u⁰ (w0 - P) - pq (p (1 - q)+r) u⁰ (w0 - P - L + c)
+p (1 - q)(1 - p (1 - q) - r) u⁰ (w0 - P - L + b)

The second derivative is negative and expected utility is thus globally concave in b for all levels of c. The
FOC thus determines the unique global maximum b* = b* (c). Evaluating the first derivative at b = L yields

^dE [uJw)] Ib=L = -ru0 (wo - P) + pq (p (1 - q) + r) (u0 (wo - P) - u0 (wo - P - L + c)) < 0
∂b

for all c ∈ [0, L], i.e. b* (c) < L for all c ∈ [0, L] and therefore b* < L.   Evaluating the first derivative at

b = 0 yields

∂E [u (w)] I         - (1 - p) (p (1 - q) + r) u⁰ (w₀ - P) - pq (p (1 - q) + r) u⁰ (w₀ - P - L + c)

∂b    ^|b=0 =                  +p (1 - q) (1 - p (1 - q) - r) u⁰ (wo - P - L)                  ^.

For any r ≥ 1 - p (1 - q), we have

∂E [u (w)],

∂b    ^lb=⁰ < ⁰

for all c ∈ [0, L], i.e. b^* (c)=0for all r ≥ 1 - p (1 - q).Forr =0we get

^{dE [u (w)]} I       = Γ (1 - p) p (1 - q) (u⁰ (wo - P - L) - U⁰ (wo - P))

∂b   Ib=^=     +pqp (1 - q) (u⁰ (wo - P - L) - u⁰ (wo - P - L + c))

¹³ There are other benefits from profit based contingent commissions. If brokers have better information about p olicyholder
risk than insurers, brokers can send a signal to insurers ab out risk type and thereby mitigate adverse selection. Profit sharing
with insurers will render the signal credible. See David J. Cummins and Neil A. Doherty, The Economics of Insurance
Intermediaries, American Insurance Association, 2005.

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