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I encounter the following problem today. It seems a simple question.

Let $U$ be a real function from $R^+\rightarrow \bar{R}$ satisfying the following conditions:

(1) $U$ is concave, continuous, and strictly increasing,

(2) $\limsup_{x\rightarrow +\infty}\frac{xU^{'}(x)}{U(x)} <1.$

(3) $ U^{'}(0+) = +\infty, \mbox{ and }U^{'}(+\infty) = 0.$

Is the following statement true?

$\bf Claim:$ For any non-negative random variable $\xi,$ if $E[U(\xi)] < +\infty,$ then we have $$E[\xi U^{'}(\xi)] < +\infty.$$

$\bf Remark:$ If $U(0) >-\infty,$ it is trivial if one notices that $$0\leq \xi U^{'}(\xi) \leq U(\xi) - U(0).$$ But in general, the property $U(0) >-\infty$ does NOT hold. For example $\ln(x).$

Any comment and suggestion are welcome. Thanks.

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As you may assume wlog '$U(\infty)>0$', it should follow directly from the asymptotic elasicity condition (2), no? As this inequality holds for $x$ sufficiently large, you can just separate the domain of integration and use for the second part the inequality... – Stephan Sturm Jun 5 '12 at 20:45
Yes. If $x$ is large, there is no problem. But problem arises when $x$ goes to zero such as $\ln(x).$ – Jun Jun 5 '12 at 21:20
up vote 2 down vote accepted

Take $U(x) = -e^{1/x^2}$ and the probability density function $f(x) = C e^{-1/x^2}$, $x \in [0;1]$, where $C$ is normalization constant.

Then, $U'(x) = \frac{2 e^{1/x^2}}{x^3}$, $E(U(\xi)) = \int_0^1 U(x) f(x) dx = -C$, $E(\xi U'(\xi)) = \int_0^1 x U'(x) f(x) dx = 2C \int_0^1 \frac{dx}{x^2} = +\infty$

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Thanks for your elegant counterexample. – Jun Jun 6 '12 at 1:40

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