rent provides “hold up” power to the policyholder. Suppose a non-verifiable loss occurs. The size of the
loss is L, but the policyholder believes (for reasons we will examine presently) it would be appropriate for
the insurer to make a non-contractual payment of b to the policyholder. Would the insurer be willing to
make a non-contractual payment for this non-verifiable loss in order to retain the customer and secure these
future rents?
We start by assuming that the policyholder has a prior expectation that the insurer will indeed make
a payment of b for a non-verifiable loss. At inception, the policyholder has the same prior for any insurer
from whom it might write the contract, but the policyholder selects one, the incumbent. All insurers charge
the same premium. If a non-verifiable loss arises and the insurer makes a payment, the policyholder’s
prior for such payment in the future from this insurer is revised upwards. Thus, the posterior that the
generous incumbent will make such settlements in the future is higher than for potential rivals. Given that
all insurers charge the same premium going forward, the policyholder chooses to renew the policy with the
incumbent. However, if the insurer does not pay the non-verifiable loss, the policyholder revises the prior
downwards. Thus, the posterior for the incumbent is below the prior for rivals and, given the same premium,
the policyholder will switch to a rival.
Any possible subgame perfect equilibrium to this game must involve either
1. insurers charge a premium in excess of the expected value of verifiable losses, P>pqc, and will choose
to make a payment, b, should a non-verifiable loss occur. To ensure that expectations are met in
equilibrium, the prior that such payment will be made is one; or
2. insurers charge a premium which just covers the expected value of the verifiable loss, P = pqc. For
these insurers, the prior is zero.
Consider now a steady state in which, in any future year, policyholders might have an expectation that
a payment of b will be made against the unverifiable loss. In this case, future rents are reduced by the
expectations of future “type-b” payment. Thus, assuming b to be constant over time, the insurer will make
the payment, b,onlyifb is less than future rents discounted at the interest rate r, i.e. if
b≤
P - pqc - p (1 - q) b
If the policyholder has all the bargaining power, then the premium P which includes future “type-b”
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