The name is absent

in a bilateral relationship, b* and c*. We observe that difference is based on a pure price effect. The overall
implicit loading r is strictly lower than in the bilateral case if and only if

(1 + γ)p (1 - q) < 1.

We thus have to examine the comparative statics of the transfer b and coverage c under changes of the
implicit loading. Insurance can be a Giffen good under decreasing absolute risk aversion if the income
effect outweighs the substitution effect. To abstract away from those wealth effects, we assume that the
policyholder’s preferences exhibit constant absolute risk aversion (CARA). Under this assumption, the
following proposition shows that coverage for both verifiable and non-verifiable losses can be obtained more
efficiently through the broker than under the bilateral case.

Proposition 3 Suppose that policyholder’s preferences exhibit CARA and that (1 + γ) p (1 - q) < 1. Then
rbr > r.  Furthermore, b*_r > b* and c_br > c* for all r < rb_r and b*_r = b* = 0 and c_br = c* for all r ≥ rbr.

Proof. See Appendix A.2. ■

These results show that the brokers can play an important coordinating mechanism in securing the
incomplete contract. In particular, by pooling risk and using the hold up power form their whole book of
business, the broker can secure implicit coverage for the non-verifiable loss on more advantageous terms and
this will lead to a higher values of b^* and c^* .

5 Summary and Comments on Contingent Commissions

The propositions are summarized in Figure 1. Without brokers, a transfer payment of the non verifiable
loss, b^* , is negotiated but it is bounded by the future rents on the policy which might be small. The present
value of these rents is of course interest rate sensitive, and the transfers are diminishing in the interest rate
and eventually disappear. These rents are bestowed on the insurer (even under competition) to create the
hold-up required for the ex post transfer for non verifiable losses. The downside to the rents is that the
insurance on non-verifiable losses is actuarially unfair and therefore only partially insured, b^* <L.This
creates a negative spill-over effect on insurance of verifiable losses. To reduce the gap in marginal utilities,
it is optimal to also partially insure verifiable losses, c^* <L. As explicit insurance is relatively cheaper than

16

<<   15   16   17   18   19   20   21   >>

More intriguing information