Arrow's Theorem of the Deductible with Heterogeneous Beliefs

The North American Actuarial Journal, 21(1):15-35, 2017

27 Pages Posted: 30 Mar 2012 Last revised: 16 Mar 2017

Date Written: March 12, 2015


In Arrow’s classical problem of demand for insurance indemnity schedules, it is well-known that the optimal insurance indemnification for an insurance buyer - or decision maker (DM) - is a deductible contract, when the insurer is a risk-neutral Expected-Utility (EU) maximizer and when the DM is a risk-averse EU-maximizer. In Arrow’s framework, however, both parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. This paper re-examines Arrow’s problem in a setting where the DM and the insurer have different subjective beliefs. Under a requirement of compatibility between the insurer’s and the DM’s subjective beliefs, we show the existence and monotonicity of optimal indemnity schedules for the DM. The belief compatibility condition is shown to be a weakening of the assumption of a monotone likelihood ratio. In the latter case, we show that the optimal indemnity schedule is a variable deductible schedule, with a state-contingent deductible that depends on the state of the world only through the likelihood ratio. Arrow’s classical result is then obtained as a special case.

Keywords: Optimal insurance, deductible contract, subjective probability, heterogeneous beliefs.

JEL Classification: G22, C02, D86

Suggested Citation

Ghossoub, Mario, Arrow's Theorem of the Deductible with Heterogeneous Beliefs (March 12, 2015). The North American Actuarial Journal, 21(1):15-35, 2017. Available at SSRN: or

Mario Ghossoub (Contact Author)

University of Waterloo ( email )

Dept. of Statistics & Actuarial Science
200 University Ave. W.
Waterloo, Ontario N2L 3G1


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