- Utility functions and their equivalence
- Risk Aversion.
- CRRA utility and its equivalence with Iso-elastic utilities.
A utility function is used to quantify the value that you gain from an outcome
.
Def 1 [Utility Function] For , a utility function is a function
that is increasing, i.e. if
component-wise then
. The utility of a random variable
is then its expected utility,
. A utility function creates an ordering where an outcome
is preferred to
if
.
Jensen inequality applies to a concave utility:
So we prefer a certain outcome rather than the risky outcome
that has the same mean – This is being risk averse.
Def 2 [Risk Aversion] If the function is concave then we also say that the function is risk averse. (Unless stated otherwise we assume that the utility function is risk averse).
Def 3 [Equivalent Utilities] We say that two utility functions and
are equivalent if they induce the same ordering. I.e.
iff $latex\mathbb{E} V(X) \leq \mathbb{E} V(Y)$.
Ex 1 Show that two utility functions are equivalent iff the same as
up-to an affine transform, i.e.
for constants and
.
CRRA Utilities
Def 4 [Coefficient of Relative Risk Aversion] For a utility function (twice differentiable) the Coefficient of Relative Risk Aversion is
Ex 2 You have utility function . You are offered a bet that increases you wealth
multiplicatively by
here
is a “small” positive is a RV. Discuss why you would accept the bet iff
I.e. You accept the bet if you mean is large but a large variance makes this less likely, and the coefficient of relative risk aversion decides the threshold.
Def 5 [CRRA Utility/ [Util:CRRA] A Constant Relative Risk Averse utility (CRRA) takes the form
Def 6 [Iso-elastic Utilities] A utility function is Iso-elastic if it is unchanged under multiplication: for all ,
I.e. the utility only cares about the relative magnitude of the risk.
Ex 3 Show that a utility function is iso-elastic iff it is a CRRA utility (up-to an affine transform).
Answers
Ans 1 Define s.t.
. Let
w.p. and
w.p.
. Then
This implies is linear.
Ans 2. Accept if
Ans 3 By [1], its immediate that CRRA implies isolastic. Further by [1], ,
for constants
and
. Differentiate twice w.r.t.
and divide gives
Set and integrate twice w.r.t.
gives the required result.