Utility Theory

  • 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:

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.

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

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

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 c>0,

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

This implies \phi is linear.

Ans 2. Accept if

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

Set x=1 and integrate twice w.r.t. c gives the required result.

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