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

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?

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!

• 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...? Mar 24 at 12:20
• @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.
– EconJohn
Mar 24 at 17:41
• Hmmm..., I'm sure it does not exist. After all, that's what my proof says. Mar 25 at 19:23
• 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. Mar 25 at 19:55
• @VARulle you are correct on both fronts. I'll edit my answer.
– EconJohn
Mar 26 at 2:23