- 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.