Consider an atemporal setting in which an individual has a utility function u of consumption. His current consumption is

c. As always, the absolute risk aversion is ARA

(c) = −u(c)/u

(c) and the relative risk aversion is RRA

(c) = −cu(c)/u

(c).

Let ε ∈ [0,c] and consider an additive gamble where the individual will end up with a consumption of either c + ε or c − ε. Define the additive indifference probability π

(c, ε) for this gamble by u

(c) =

1 2 + π

(c, ε)

u(c + ε) +

1 2 − π

(c, ε)

u(c − ε). (*)

Assume that π

(c, ε) is twice differentiable in ε.

(a) Argue that π

(c, ε) ≥ 0 if the individual is risk-averse.

(b) Show that the absolute risk aversion is related to the additive indifference probability by the following relation ARA

(c) = 4 lim

ε→0

∂π

(c, ε)

∂ε

and interpret this result. Hint: differentiate twice with respect to ε in (*) and let ε → 0.

Now consider a multiplicative gamble where the individual will end up with a consumption of either (1 + ε)c or (1 − ε)c, where ε ∈ [0, 1]. Define the multiplicative indifference probability 

(c, ε) for this gamble by u

(c) =

1 2 + 

(c, ε)

u ((1 + ε)c) +

1 2 − 

(c, ε)

u ((1 − ε)c).

Assume that 

(c, ε) is twice differentiable in ε.

(c) Derive a relation between the relative risk aversion RRA

(c) and limε→0 ∂(c,ε)

∂ε and interpret the result.