<p>I am working on a model that assumes expected utility maximisation with a vNM function <span class="math-container">$u$</span> that exhibits some standard properties:</p>
<ol>
<li><span class="math-container">$u'(w) &gt; 0$</span> and <span class="math-container">$u''(w) &lt; 0$</span> for all <span class="math-container">$w \geq 0$</span>.</li>
<li><span class="math-container">$\lim_{w \rightarrow 0^+}u'(w) = \infty$</span>.</li>
<li><span class="math-container">$-u''(w)/u'(w)$</span> is strictly decreasing in <span class="math-container">$w$</span> over <span class="math-container">$w \geq 0$</span>.</li>
</ol>
<p>As a leading example, I have chosen <span class="math-container">$u(w) = \ln(w)$</span>. However, I now wonder if this satisfies (1) or (3) given that <span class="math-container">$\ln(w)$</span>, and its derivative <span class="math-container">$1/w$</span>, are both undefined when <span class="math-container">$w = 0$</span>.</p>
<p>Relatedly, it is hard for me to argue that, in the optimal lottery, wealth is positive in every state: I <em>want</em> to say that <span class="math-container">$\ln(0) = -\infty$</span>, which cannot be optimal, but this does not seem rigorous.</p>
<p>Assuming this is a problem, is there a simple solution?</p>

CC BY-SA 4.0
2024-07-30T15:30:57.223Z
How about $u(w)=\ln(1+w)$? It is continuous and well-defined at $0$.
Amit
CC BY-SA 4.0
2024-07-30T20:00:57.193Z
Why is it not rigorous to extend the function so that $u(0)=-\infty$?
Michael Greinecker
CC BY-SA 4.0
2024-08-02T17:27:23.420Z
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.
Joseph Basford
CC BY-SA 4.0
2024-08-02T23:27:57.407Z
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
CC BY-SA 4.0
2024-08-05T16:17:15.187Z
Thanks for the suggestions everyone; I'll need to think a bit more about this
afreelunch
<microeconomics><risk>
Logarithmic utility and zero consumption
afreelunch
# Logarithmic utility and zero consumption

(Asked by: username_0 on 2024-07-30)

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?

**Tags:** microeconomics, risk
> **Comments:**
> **username_1**: How about $u(w)=\ln(1+w)$? It is continuous and well-defined at $0$.  
> **username_2**: Why is it not rigorous to extend the function so that $u(0)=-\infty$?  
> **username_3**: 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.  
> **username_1**: How about $u(w)=\sqrt{w}$? This function satisfy the desired properties 1 and 3 for all $w>0$, and also satisfy property 2.  
> **username_0**: Thanks for the suggestions everyone; I'll need to think a bit more about this  

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## 0 answers

