Editing Expected utility hypothesis (section)

==== Examples of von Neumann–Morgenstern utility functions ====
The utility function <math>u(w)=\log(w)</math> was originally suggested by Bernoulli (see above). It has [[relative risk aversion]] constant and equal to one and is still sometimes assumed in economic analyses. The utility function

:<math> u(w)= -e^{-aw}</math>

It exhibits constant absolute risk aversion and, for this reason, is often avoided, although it has the advantage of offering substantial mathematical tractability when asset returns are normally distributed. Note that, as per the affine transformation property alluded to above, the utility function <math>K-e^{-aw}</math> gives the same preferences orderings as does  <math>-e^{-aw}</math>; thus it is irrelevant that the values of <math>-e^{-aw}</math> and its expected value are always negative: what matters for preference ordering is which of two gambles gives the higher expected utility, not the numerical values of those expected utilities.

The class of constant relative risk aversion utility functions contains three categories. Bernoulli's utility function

:<math> u(w) = \log(w)</math>

Has relative risk aversion equal to 1. The functions

:<math> u(w) = w^{\alpha}</math>

for <math>\alpha \in (0,1)</math> have relative risk aversion equal to <math>1-\alpha\in (0,1)</math>.  And the functions

:<math> u(w) = -w^{\alpha}</math>

for <math>\alpha < 0</math> have relative risk aversion equal to <math>1-\alpha >1.</math>

See also [[Risk aversion#Absolute risk aversion|the discussion]] of utility functions having hyperbolic absolute risk aversion (HARA).