There are some von-Neumann utility functions that I come across quite often in different articles / books like: $ U(x)=\ln(x)$, $U(x)= \frac {1}{\gamma}x^\gamma$ with $\gamma <1$ and $U(x)=\frac {1-\gamma}{\gamma}(\frac{\alpha x}{1-\gamma} +\beta)^{\gamma}$ subject to the restrictions $\gamma \neq 1, \frac{\alpha x}{1-\gamma} +\beta >0$, and $\beta=1$ if $\gamma=-\infty$.

My question is: Why are those functions used that often? It's not like they are the only "simple" concave and increasing functions out there?

So I guess there must be some specific properties that make them interesting?


2 Answers 2


These are a natural and easiest (most tractable mathematically) choice.

A utility function is defined up to a positive affine transformation: economically there is no difference between the utility functions $U(x)$ and $\tilde{U}(x)=Au(x)+B$. Hence, a measure of risk aversion that remains constant w.r.t. affine transformations would be useful. How does one construct such a measure? Well, the easiest way is to consider the expression $$A(x)= -\frac{U''(x)}{U'(x)}$$ a.k.a. ARA (Arrow–Pratt measure of absolute risk aversion). ARA stays the same under affine transformations and measures the degree of risk aversion - the curvature of the utility function. The reciprocal of ARA measures the level of risk tolerance, and a simple special case is when it is a linear function of wealth: $$T(x)=\frac{1}{A(x)}=\frac{x}{1-\gamma}+\frac{b}{a}.$$ Now, what are the utility functions such that the corresponding level of risk tolerance is linear? These are solutions to the ODE $$-\frac{U'(x)}{U''(x)}=\frac{x}{1-\gamma}+\frac{b}{a}$$ which is known to be solvable in closed form. The unique solution (up to affine transformation!) to the equation has the form $$\qquad U(x)=\frac{1-\gamma}{\gamma}\left(\frac{ax}{1-\gamma}+b \right)^\gamma.\qquad(1)$$ There are other solutions which differ from (1) by additive and/or multiplicative constants but these do not affect the behavior implied by the utility function. (1) is known as the hyperbolic absolute risk aversion.

The other utility functions that you've mentioned are just specifications of (1). In particular, assuming $b=0$ one gets the isoelastic utility: $$\quad\qquad U(x) = \begin{cases}\frac{x^\gamma-1}{\gamma},\quad \gamma\neq 0 \\ \ln(x), \quad \gamma =0 \end{cases}\qquad\quad (2)$$ (2) is also the only example of utility functions with the constant relative risk aversion $$R(x)=xA(x)=1-\gamma.$$


The functions are set up so that maximizing the expectation of the utility function will obey the von Neumann-Morgenstern axioms (completeness, transitivity, independence, continuity). As you've pointed out, convexity is also a desired behavior and certainly you can come up with other functions.


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