Utility functions and their equivalence
Risk Aversion.
CRRA utility and its equivalence with Iso-elastic utilities.

A utility function $U(x)$ is used to quantify the value that you gain from an outcome $x$ .

Def 1 [Utility Function] For $\mathcal{X}\subset\mathbb{R}^d$ , a utility function is a function $U:\mathcal{X} \rightarrow \mathbb{R}$ that is increasing, i.e. if $x\leq y$ component-wise then $U(x)\leq U(y)$ . The utility of a random variable $X$ is then its expected utility, $\mathbb{E} U(X)$ . A utility function creates an ordering where an outcome $X$ is preferred to $Y$ if $\mathbb{E} U(X) \geq \mathbb{E} U(Y)$ .

Jensen inequality applies to a concave utility:

$\mathbb{E} U(X) \leq U(\mathbb{E} X)$

So we prefer a certain outcome $\mathbb{E} X$ rather than the risky outcome $X$ 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 $U$ and $V$ are equivalent if they induce the same ordering. I.e. $\mathbb{E} U(X) \leq \mathbb{E} U(Y)$ iff $latex\mathbb{E} V(X) \leq \mathbb{E} V(Y)$.

Ex 1 Show that two utility functions are equivalent iff $V$ the same as $U$ up-to an affine transform, i.e.

$\begin{aligned} V(x) = a U(x) +b \end{aligned}$

for constants $a>0$ and $b$ .

CRRA Utilities

Def 4 [Coefficient of Relative Risk Aversion] For a utility function $U:\mathbb{R} \rightarrow\mathbb{R}$ (twice differentiable) the Coefficient of Relative Risk Aversion is

$- x\frac{U''(x)}{U'(x)}.$

Ex 2 You have utility function $U$ . You are offered a bet that increases you wealth $w$ multiplicatively by $(1+ X)$ here $X$ is a “small” positive is a RV. Discuss why you would accept the bet iff

$\frac{2\bE X}{\bE X^2} \geq - x\frac{U''(x)}{U'(x)}$

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

$U(x) = \begin{cases} \frac{x^{1-R}}{1-R}, & R \neq 1, \\ \log x, & R=1. \end{cases}$

Def 6 [Iso-elastic Utilities] A utility function is Iso-elastic if it is unchanged under multiplication: for all $c>0$ ,

$\bE U(X) \geq \bE U(Y) \qquad \text{iff} \qquad \bE U(cX) \geq \bE U(cY).$

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 $\phi: \mathbb{R} \rightarrow\mathbb{R}$ s.t. $\phi(\mathbb{E} U(X))=\mathbb{E} V(X)$ . Let $X= x$ w.p. and $X=y$ w.p. $q=1-p$ . Then

$\phi(pU(x)+qU(y))= p V(x) +qV(y)=p\phi(U(x)) + (1-p) \phi(U(y)).$

This implies $\phi$ is linear.

Ans 2. Accept if $0 \leq \bE [U(w(1+ X)) - U(w)] \stackrel{\tiny{Taylor}}{\approx} \bE \left[U'(w) w X + \frac{1}{2} U''(w) w^2 X^2 \right].$

Ans 3 By [1], its immediate that CRRA implies isolastic. Further by [1], $\forall c$ , $U(cx)= a_c U(x) + b_c$ for constants $a_c$ and $b_c$ . Differentiate twice w.r.t. $x$ and divide gives