The name is absent

payment is given by

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

The premium is the sum of the actuarially fair value, pqc + p (1 - q) b, and a loading, rb. This loading
represents the future rents that provides the insurer with the incentive to pay the transfer b in case of a
unverifiable loss.

Would the policyholder and insurer wish to engage in this partially incomplete contract in which there
is conventional insurance coverage together with the ability of the policyholder to hold-up the insurer to
pay non verifiable, and therefore not contracted losses? If so, then such an arrangement provides explicit
insurance for contractible losses and implicit insurance for the non-verifiable (non-contractible) losses.

Proposition 1 In the model outlined above, 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 the non-verifiable event,
i.e. b* < c* < L for all r. Furthermore, there exists a critical discount rate r > 0 such that it is optimal
to generate a transfer if the discount rate is below r and not to generate a transfer otherwise, i.e. b* > 0 if
r < r and b* = 0 if r ≥ r.

Proof. See Appendix A.1. ■

For r <r,we thus have a subgame perfect equilibrium where insurers charge a premium in excess of the
expected value of verifiable losses and will choose to make a payment for a non-verifiable loss. Policyholders
have a prior of one that such payment will be made. For all r ≥ r, we have a subgame perfect equilibrium
where only verifiable events are covered and policyholders have a prior of zero that payments for non-verifiable
losses are made.

The intuition behind these results is as follows. In order to generate a “hold-up” and thereby payments
for unverifiable losses, the policyholder has to pay a loading rb. The premium is therefore unfair and full
“insurance” of the unverifiable loss is not optimal.¹¹ Purchasing full coverage of the verifiable loss would
then lead to a higher marginal utility in the state of the unverifiable event. The policyholder thus finds
it optimal to transfer wealth into that state by not buying full coverage of the verifiable loss and thereby
reducing the premium. Last, the discount rate measures the level of the loading, rb, and for high discount
rates the policyholder will find it optimal to not generate the transfer b.

¹¹ The result still holds if there is only a non-verifiable loss, i.e. if q =0, and is therefore robust with respect to different
correlation structures between the verifiable and unverifiable loss.

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