Consider an agent who behaves according to the von Neumann- Morgenstern axioms with Bernoulli utility function u : R → R satisfying
u′ > 0 > u′′. Denote by
R(y) = −u′′ (y) / u′ (y)
the agent’s Arrow-Pratt coefficient of absolute risk aversion. The agent exhibits constant absolute risk aversion (CARA) if R(y) is constant for all y ∈ R, and exhibits non-increasing absolute risk aversion (NIARA) if R (y) is non-increasing in y.
Let Y be a real-valued lottery with cumulative distribution F (.) and density f (.), and denote by mY its certainty equivalent. Show that if the agent exhibits CARA then
∫u′ (y) dF (y) = u′ (mY )