<p>Prudence has to do with the response of how <em>additional dimensions of uncertainty</em> impacts the preference or aversion to that uncertainty.</p>
<p>To illustrate, prudence (the sign of <span class="math-container">$u'''(\cdot)$</span>) impacts the preference that is had between two expected utilities:
<span class="math-container">$$\mathbb{E}[U_1]=pu(qx_1+(1-q)x_2)+(1-p)u(ry_1+(1-r)y_2)$$</span>
and
<span class="math-container">$$\mathbb{E}[U_2]=p(qu(x_1)+(1-q)u(x_2))+(1-p)(ru(y_1)+(1-r)u(y_2))$$</span></p>
<p>Those who are prudent <span class="math-container">$u'''(\cdot)&gt;0$</span> would prefer <span class="math-container">$\mathbb{E}[U_1]$</span> to <span class="math-container">$\mathbb{E}[U_2]$</span> while those who are imprudent <span class="math-container">$u'''(\cdot)&lt;0$</span> would prefer <span class="math-container">$\mathbb{E}[U_2]$</span> to <span class="math-container">$\mathbb{E}[U_1]$</span>.</p>
<p>Prudence has everything to do with the response to additional uncertainty and thats the key message in static enviroments.</p>
<p>NOTE: If this answer isn't clear, Id recommend reading chapter 3 of <a href="https://books.google.ca/books?hl=en&amp;lr=&amp;id=RJY4Q0O-R1AC&amp;oi=fnd&amp;pg=PA6&amp;dq=louis%20eekhoudt%20risk%20and%20medical%20decision%20making&amp;ots=PqdAf_itmP&amp;sig=nRXRKXCrSgq6-H0yMKjyq2jRBJM#v=onepage&amp;q=louis%20eekhoudt%20risk%20and%20medical%20decision%20making&amp;f=false" rel="nofollow noreferrer">Risk and Medical Decision Making </a> by Louis Eekhoudt. while health may not be your field, I have found this text helpful for understanding the concept.</p>
<p>I hope this helps!</p>

CC BY-SA 4.0
2023-03-24T12:20:47.070Z
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...?
VARulle
CC BY-SA 4.0
2023-03-24T17:41:07.747Z
@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
CC BY-SA 4.0
2023-03-25T19:23:35.670Z
Hmmm..., I'm sure it does not exist. After all, that's what my proof says.
VARulle
CC BY-SA 4.0
2023-03-25T19:55:49.010Z
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.
VARulle
CC BY-SA 4.0
2023-03-26T02:23:00.810Z
@VARulle you are correct on both fronts. I'll edit my answer.
EconJohn
CC BY-SA 4.0
2023-03-23T17:23:34.190Z
2023-03-26T02:24:44.600Z
2023-03-26T02:24:44.600Z
EconJohn
<p>How can we interpret a &quot;prudent&quot; agent in the static case (i.e., someone with <span class="math-container">$u'''(\cdot)&gt;0$</span>)?</p>
<p>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 <a href="https://economics.stackexchange.com/questions/20765/intuition-behind-the-definition-of-prudence">here</a>). Is it the only case where we have an interpretation of the third-order derivative of the utility function?</p>
<p>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:</p>
<p><span class="math-container">$$-\frac{u'''(x)}{u''(x)}\geq-\frac{u''(x)}{u'(x)}$$</span></p>
<p>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?</p>

<macroeconomics><microeconomics><utility><decision-theory><risk-aversion>
What is the economic intuition of prudence in the static case?
ju_pi_car
# What is the economic intuition of prudence in the static case?

(Asked by: username_0 on 2023-03-23)

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](https://economics.stackexchange.com/questions/20765/intuition-behind-the-definition-of-prudence)). 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?

**Tags:** macroeconomics, microeconomics, utility, decision-theory, risk-aversion


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## 1 answers

### Answer by username_1 on 2023-03-23. Score: 4 | Accepted answer

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](https://books.google.ca/books?hl=en&lr=&id=RJY4Q0O-R1AC&oi=fnd&pg=PA6&dq=louis%20eekhoudt%20risk%20and%20medical%20decision%20making&ots=PqdAf_itmP&sig=nRXRKXCrSgq6-H0yMKjyq2jRBJM#v=onepage&q=louis%20eekhoudt%20risk%20and%20medical%20decision%20making&f=false)  by Louis Eekhoudt. while health may not be your field, I have found this text helpful for understanding the concept.

I hope this helps!

> **Comments:**
> **username_2**: 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...?  
> **username_1**: @username_2 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.  
> **username_2**: Hmmm..., I'm sure it does not exist. After all, that's what my proof says.  
> **username_2**: 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.  
> **username_1**: @username_2 you are correct on both fronts. I'll edit my answer.  

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