Page 30 - 36-1

Ambiguity Increases and Insurance Deductibles

by Gollier (2014) under the smooth ambiguity aversion model. Gollier (2014) finds that
the optimal insurance contract under ambiguity aversion is the same as under ambiguity
*
neutrality when the ambiguity only affects loses above D .
Next, we investigate when a risk- and ambiguity-averse individual facing a specific

ambiguity increase lowers the optimal deductible (i.e., increases insurance coverage), as
this matters to both the individual and the insurer. We obtain the determining condition
stated in Case 2 of Proposition 4 when relaxing Assumption 2 by allowing for Δ with a
T
*
different center from Δ while keeping G(x;π ) ∈ Δ ⊂ Δ . We refer to this as a nonspecific
T
F
F
ambiguity increase. Under this type of ambiguity increase, the individual chooses a lower
optimal deductible than D when the worst loss distribution remains unaffected while the
*
F
best loss distribution (and the loss distribution distorted by α) deteriorates in the sense
*
of FSD for losses below D . The intuition is as follows. An extremely ambiguity-averse
F
individual (α = 1) who pessimistically believes the worst loss distribution will be realized
*
*
still chooses D since the uncovered loss x ∈ [0,D ] is unaffected by the nonspecific
F
F
ambiguity increase. On the other hand, a non-extremely ambiguity-averse individual (α
1
∈ ( ,1) ) believes that the realized loss distribution could be the worst or best one. The
2
worst distribution remains unchanged, whereas under the best distribution, the mean loss
*
is reduced. This means that at D , the individual is now more likely to take the full loss,
F
without any indemnity from the insurer. Based on FOC (8), after the nonspecific ambiguity
*
increase, lowering the deductible level at D reduces the expected marginal utility cost in
F
the uncovered loss state while leaving the net marginal utility benefit covered by the partial
indemnity unchanged. Therefore, it is optimal for the individual to lower the deductible.
We demonstrate Proposition 4 in Figure 5 through a numerical example of a specific

ambiguity increase. The full settings of the example are reported in Appendix C, Table
1. To show the key results clearly, we zoom in on the point where the individual starts
to receive positive indemnity, indicating the optimal deductible level. In this example,
in the absence of ambiguity, D = 2.01 (the dark dotted line) and introducing ambiguity
*
*
lowers the optimal deductible to D = 2 (the dark solid line). The loss distributions after
F
the specific ambiguity increase, G(x;π ) and G(x;¯π ) satisfy Assumptions 2 and 3 and Case
T
_ T
1 of Proposition 4 (Panel C of Table 1). Thus, after the specific ambiguity increase, the
optimal deductible level does not change (D = 2, the dark solid line).
*
T
Let us consider another example. Here, a nonspecific ambiguity increase occurs, and

22