The name is absent

where the right hand side represents the future rents for the insurer from retaining the entire book of business.

Assuming policyholders have all the bargaining power, the net premium paid to the insurer is then

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

We observe that the implicit loading, p (1 - q) rb, is smaller than the implicit loading that is needed in
the bilateral relationship without the broker, rb. Ex-ante the policyholder chooses the level of insurance
coverage, c, of the verifiable loss and the transfer, b, of the non-verifiable loss to maximize expected utility
subject to the premium structure above and the compensation paid to the broker. The compensation
structure is linear in the insurer’s expected profits, i.e.

k = γp(1 - q) rb.

The overall payment for each policyholder can then be written as

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

Defining

r = (1 + γ) p (1 — q) r

yields

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P = P + k = pqc + p (1 — q) b + rb.

The premium structure is thus the same as in the bilateral case which leads to results that are qualitatively
equivalent to Proposition 1.

Proposition 2 In the market with the broker, it is optimal for the policyholder to partially insure both types
of losses and to buy more coverage on the verifiable than to implicitly generate on non-verifiable event, i.e.
b£_r < cbr < L for all r. Furthermore, there exists a critical discount rate r_br > 0 such that it is optimal to
generate a transfer if the discount rate is below r_br and not to generate a transfer otherwise, i.e. b^t_br > 0 if
r < rb_r and b_br = 0 if r ≥ rb_r.

Next, we compare the optimal transfer bb*_r and coverage c*_br generated through the broker with the ones

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