bbr for non verifiable. Brokers can advise policyholders to move that business and thereby extract a penalty
from the incumbent insurer for failure to make an appropriate payment for a non-verifiable loss. Each
policyholder observes its settlement of a non-verifiable claim, or the broker’s penalty on the insurer if it is
not paid. Since each insurer has portfolio with each broker, the broker’s reputation for enforcing (failing to
enforce) penalty quickly becomes known to all insurers. If the broker is known not to apply a penalty, then
the insurer makes future payment of bbr =0.
Initially, each policyholder has a uniform prior that the incumbent broker, and rival brokers, will use
its leverage over insurers to encourage settlement of non verifiable losses.12 If the insurer fails to make a
settlement and the incumbent broker then fails to extract a penalty from the insurer, the policyholder revises
the prior downwards. This posterior is now lower than the common prior for rival brokers and the consumer
switches brokers.
Now some proportion of these policyholders will suffer an non verifiable loss and the policyholders will
have an expectations that the broker will facilitate a settlement of bbr. The value bbr is limited by the rents
and the broker is expected to impose a penalty such that the insurer will weakly better off settling at this
value. We have estimated the penalty under the assumption that all ex post bargaining power lodges with
the policyholder. This is not necessary for the general structure of our results, but it permits considerable
simplification. If the insurer fails to make this settlement and the broker fails to extract the penalty, the
affected policyholders cancel their business and switch to another broker. Because the brokers compensation
increases with the number of clients, then the broker has an incentive to present the penalties to the insurer
and enforce them if the settlement is not reached. Anticipating the ex post disposition of the non-verifiable
claims, the policyholder’s ex ante problem is to secure, with the help of the broker, an incomplete contract
with explicit and implicit coverage of c*br and b*br.
Now suppose that the size of the book of business, m, is large such that we can approximate the number of
non-verifiable losses occurring in each period by the expected number, mp (1 - q). The incentive constraint
for the insurer to pay the transfer b is then
P -pqc-p(1 - q)b
mp (1 — q) b ≤ m----------------
r
12 Recall that we have assumed all non verifiable losses to be ex-post insurable. If there also were ex-post uninsurable
losses, the broker would be called to use its judgement to distinguish and to use its hold-up power only for ex-post insurable
losses. Further structure could be added to our model by allowing brokers to penalize policyholders who tried to falsify ex-post
uninsurable claims to appear as though they were insurable.
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