What is the economic intuition of prudence in the static case?

Question

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?

Answer 1

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!