# Logarithmic utility and zero consumption

I am working on a model that assumes expected utility maximisation with a vNM function $$u$$ that exhibits some standard properties:

1. $$u'(w) > 0$$ and $$u''(w) < 0$$ for all $$w \geq 0$$.
2. $$\lim_{w \rightarrow 0^+}u'(w) = \infty$$.
3. $$-u''(w)/u'(w)$$ is strictly decreasing in $$w$$ over $$w \geq 0$$.

As a leading example, I have chosen $$u(w) = \ln(w)$$. However, I now wonder if this satisfies (1) or (3) given that $$\ln(w)$$, and its derivative $$1/w$$, are both undefined when $$w = 0$$.

Relatedly, it is hard for me to argue that, in the optimal lottery, wealth is positive in every state: I want to say that $$\ln(0) = -\infty$$, which cannot be optimal, but this does not seem rigorous.

Assuming this is a problem, is there a simple solution?

• How about $u(w)=\ln(1+w)$? It is continuous and well-defined at $0$.
– Amit
Commented Jul 30 at 15:30
• Why is it not rigorous to extend the function so that $u(0)=-\infty$? Commented Jul 30 at 20:00
• It may be unrelated to your point, but one issue you may run in to is $\infty - \infty$, which is not well defined a priori. Consider the lottery that with $1/2$ probability gives $w=0$ and gives $w=\exp\{\frac{\pi^2}{3}n^2\}$ with probability $\frac{3}{\pi^2} \frac{1}{n^2}$ for $n\in \mathbb{N}$. The expected utility of such a lottery is $\frac{1}{2}u(0)+\sum_{n=1}^{\infty}\frac{3}{\pi^2} \frac{1}{n^2}u(\exp\{\frac{\pi^2}{3}n^2\})=\infty-\infty$. One solution is to define this $\infty-\infty$ to be something. Another is to restrict attention to lotteries with bounded wealth outcomes. Commented Aug 2 at 17:27
• How about $u(w)=\sqrt{w}$? This function satisfy the desired properties 1 and 3 for all $w>0$, and also satisfy property 2.
– Amit
Commented Aug 2 at 23:27
• Thanks for the suggestions everyone; I'll need to think a bit more about this Commented Aug 5 at 16:17