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How can we interpret a "prudent" agent in the static case (i.e., someone with $u'''(\cdot)>0$)?

I understand that in a dynamic setting, someone exhibiting prudence would do precautionary savings to face future risky situations (see a post on this specific question here). Is it the only case where we have an interpretation of the third-order derivative of the utility function?

I ask this question because a necessary and sufficient condition for a utility function to exhibit decreasing absolute risk aversion is to have prudence uniformly larger than absolute risk aversion:

$$-\frac{u'''(x)}{u''(x)}\geq-\frac{u''(x)}{u'(x)}$$

Absolute risk aversion is quite intuitive to understand in a static case (the larger it is, the higher the payoff I ask for taking a small risk), what about prudence?

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Prudence has to do with the response of how additional dimensions of uncertainty impacts the preference or aversion to that uncertainty.

To illustrate, prudence (the sign of $u'''(\cdot)$) impacts the preference that is had between two expected utilities: $$\mathbb{E}[U_1]=pu(qx_1+(1-q)x_2)+(1-p)u(ry_1+(1-r)y_2)$$ and $$\mathbb{E}[U_2]=p(qu(x_1)+(1-q)u(x_2))+(1-p)(ru(y_1)+(1-r)u(y_2))$$

Those who are prudent $u'''(\cdot)>0$ would prefer $\mathbb{E}[U_1]$ to $\mathbb{E}[U_2]$ while those who are imprudent $u'''(\cdot)<0$ would prefer $\mathbb{E}[U_2]$ to $\mathbb{E}[U_1]$.

Prudence has everything to do with the response to additional uncertainty and thats the key message in static enviroments.

NOTE: If this answer isn't clear, Id recommend reading chapter 3 of Risk and Medical Decision Making by Louis Eekhoudt. while health may not be your field, I have found this text helpful for understanding the concept.

I hope this helps!

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  • $\begingroup$ Can a risk averse person be imprudent? Risk averse and imprudent means positive, decreasing, and concave marginal utility, right? But decreasing and concave together imply eventually negative...? $\endgroup$
    – VARulle
    Mar 24, 2023 at 12:20
  • $\begingroup$ @VARulle i dont know, but it seems not for the usual cases of $U(x)=x^a$ where $a \in (0,1)$ im sure some function exists though. Just not sure if its pragmatic. $\endgroup$
    – EconJohn
    Mar 24, 2023 at 17:41
  • $\begingroup$ Hmmm..., I'm sure it does not exist. After all, that's what my proof says. $\endgroup$
    – VARulle
    Mar 25, 2023 at 19:23
  • $\begingroup$ Also, the "concavity or convexity of the second derivative of your utility function" is determined by the sign of the fourth, not the third derivative of the utility function. $\endgroup$
    – VARulle
    Mar 25, 2023 at 19:55
  • $\begingroup$ @VARulle you are correct on both fronts. I'll edit my answer. $\endgroup$
    – EconJohn
    Mar 26, 2023 at 2:23

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