Risk aversion and utility transformation: are preferences still the same?

Question

If you have two utility functions $u(\cdot), \; v(\cdot)$ such that $v(x) = f(u(x))$ for some monotonic transformation $f(\cdot)$, then $u(\cdot)$ and $ v(\cdot)$ represent the same preference relation $\succsim$. This means that utility levels are meaningless to compare among individuals: if two individuals are such that one has a utility function $u(\cdot)$ and the other has $v(\cdot)$, they have the same preferences over their consumption space.

But when dealing with utility over monetary prizes, if you have two individuals with Bernoulli utilities $u_1(\cdot)$ and $u_2(\cdot)$ such that $u_1(x) = \varphi(u_2(x))$ for some concave monotonic transformation $\varphi (\cdot)$, then the individual with utility $u_1(\cdot)$ is more risk-averse than the one with $u_2(\cdot)$ (Mas-Colell Proposition 6.C.2).

My question is: how is it possible to conciliate the fact that both have the same preference relation $\succsim$ over the consumption space (since one's utility is just an monotonic transformation of the other's) and at the same time one is more risk-averse than the other?

Answer 1

It depends on what you consider the individuals' "consumption space".

Both individuals have the same preferences over monetary prizes: They prefer more money to less money. These preferences are represented by any strictly increasing utility function on the real numbers, including both Bernoulli functions.

However, they have different preferences over lotteries over monetary prizes. These preferences are represented by (any positive monotone transformations of) their von-Neumann-Morgenstern utility functions on the probability simplex. These functions' values are expected Bernoulli utilities, see Michael Greinecker's answer for details.

Answer 2

The choice objects for expected utility theory in the context of risk aversion are not amounts of money, they are lotteries over money. Bernoulli functions are not utility functions for these choice objects, their expectations are.

If $\pi$ and $\pi'$ are lotteries (with finite support), $u$ a Bernoulli function for an expected utility representation of $\succeq$, and $f$ a strictly increasing function from the real to the reals, then we have

$$\pi\succeq\pi'$$

if and only if

$$\sum_x \pi(x) u(x)\geq \sum_x \pi'(x) u(x)$$

if and only if

$$f\bigg(\sum_x \pi(x) u(x)\bigg)\geq f\bigg(\sum_x \pi'(x) u(x)\bigg).$$