<p>Consider the following form of utility function:</p>
<p><span class="math-container">$U(x,y) = \beta (x-\alpha)^3 + y$</span>, where <span class="math-container">$\alpha,\beta &gt; 0$</span> are parameters. It is easy to check this utility function is monotone, in fact, strictly monotone.</p>
<p>For a constant utility level <span class="math-container">$\overline{U}$</span>, we get the indifference curve:</p>
<p><span class="math-container">$\overline{U} = \beta (x-\alpha)^3 + y \implies y = \overline{U} - \beta (x-\alpha)^3$</span>.</p>
<p>Differentiating <span class="math-container">$y$</span> w.r.t. <span class="math-container">$x$</span> in the IC's:</p>
<p><span class="math-container">$y' = - 3 \beta (x-\alpha)^2$</span></p>
<p><span class="math-container">$y'' = -6 \beta (x-\alpha)$</span></p>
<p>We get that the IC's are convex for <span class="math-container">$x &lt; \alpha$</span> and concave for <span class="math-container">$x &gt; \alpha$</span>.</p>
<p>Differentiating one more time,</p>
<p><span class="math-container">$y''' = -6 \beta$</span>.</p>
<p>So the IC's always have a negative third derivative.</p>
<p>Below is the indifference curve for the utility level <span class="math-container">$\overline{U} = 8$</span>, for the functional form above, with parameters <span class="math-container">$\alpha = 2, \beta = 1$</span>.</p>
<p><a href="https://i.sstatic.net/8tG0S.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/8tG0S.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-03-04T18:20:20.443Z
2023-03-04T18:34:06.490Z
2023-03-04T18:34:06.490Z
Nicolas Torres
<h3>Question</h3>
<blockquote>
<p>Is there an example of consumer preferences over consumption bundles <span class="math-container">$(x,y)\in \Bbb R^2$</span> that would be <em><strong>concave</strong></em> when <span class="math-container">$x$</span> is abundant relative to <span class="math-container">$y$</span> and <em><strong>convex</strong></em> otherwise?</p>
</blockquote>
<p>Are there known situations when this happens? I have never heard of that.</p>
<hr />
<h3>Context</h3>
<p>I've been recently thinking a lot about analyzing functions <span class="math-container">$f:\Bbb R \to \Bbb R$</span> with <a href="https://math.stackexchange.com/questions/4619836/why-are-monotonicity-and-convexity-so-intensely-studied-in-comparison-to-their-h">positive third-order derivative</a> instead of functions with positive second-order derivative (convex functions). Such functions are either entirely convex / concave, or there is a unique point (inflection point) at which concavity switches into convexity. It turns out that such functions have many nice properties, for example they have at most three roots and most one local maximum and one local minimum. Therefore I expect that if a utility function <span class="math-container">$u(x,y)$</span> had positive/negative third-order derivative along lines in the space of bundles, then it would still be easy to analyze the consumer choice problem as it is in the case of convex preferences.</p>

CC BY-SA 4.0
2023-01-25T08:42:28.843Z
What are concave preferences? Convexity of preferences and convexity of functions are very different things.
Michael Greinecker
CC BY-SA 4.0
2023-01-25T09:11:12.800Z
@MichaelGreinecker I would regard preferences to be concave iff any lower contour set is concave. If the preference is represented by a utility function $u$, then $u$ is quasi-concave. I understand that concavity of $u$ is sufficient condition for concavity of the preferences, but not necessary. Note that function property of having positive third-order derivative can be also generalized to its "quasi" (i.e. ordinal) form.
Pavel Kocourek
CC BY-SA 4.0
2023-01-25T10:02:51.403Z
Quasi-concavity of a representing utility function is equivalent to the convexity (!) of the represented preferences. I don’t know what concave sets are either.
Michael Greinecker
CC BY-SA 4.0
2023-01-25T10:16:00.433Z
@MichaelGreinecker Thanks a lot for the correction! So it is reversed: convex preferences $\Leftrightarrow$ quasi-concave utility; and concave preferences $\Leftrightarrow$ quasi-convex utility.
Pavel Kocourek
CC BY-SA 4.0
2023-01-25T10:29:32.953Z
I'm still not sure what concave preferences should be? Convex upper contour sets?
Michael Greinecker
CC BY-SA 4.0
2023-01-25T10:39:07.323Z
I just see that I wrote a completely nonsense and I apologize for that: "preferences concave iff any lower contour set is concave" I meant that the set is convex. (A set can not be concave, at least that I know:-).) And I think that after this correction it is correct. In other terms, if I'm lees happy with either bundle $a$ or $b$ than with the bundle $c$, then I'm also less happy with any convex combination of $a$ and $b$ than with bundle $c$. Does this make sense? In simple terms: Convex preferences means preferring mixtures, concave preferences means disliking mixtures. Or am I confused?
Pavel Kocourek
CC BY-SA 4.0
2023-01-25T10:54:12.237Z
You can define things like that. Convexity of preferences is a widely use assumption. I have never seen concavity of preferences.
Michael Greinecker
CC BY-SA 4.0
2023-01-25T11:44:41.370Z
@MichaelGreinecker I remember Varian made a hypothetical example of sardines and ice cream. Each one is delicious however combining them makes people worse off..
T123
CC BY-SA 4.0
2023-01-25T12:02:23.240Z
@T123 There are many reasons why non-convex preferences matter. It is not clear that this specific form is particularly useful.
Michael Greinecker
CC BY-SA 4.0
2023-01-25T12:52:47.483Z
The question would eventually have a meaning if you replace "consumer preferences" by "consumers' utility function"
Bertrand
CC BY-SA 4.0
2023-01-25T13:19:09.340Z
@Bertrand Why won't the question make sense at it is? Just for a reference, convex and concave preferences are defined here: https://en.wikipedia.org/wiki/Preference_(economics)
Pavel Kocourek
CC BY-SA 4.0
2023-01-25T14:39:26.020Z
@Pavel Kocourek. OK. I just wanted to be sure that there was no confusion (because you mentioned positive/negative third-order derivative of the utility function, which is unrelated to nonconvex preferences).
Bertrand
<consumer-theory><preferences><choice-theory>
Example of consumer preferences that switches from being concave to being convex
Pavel Kocourek
# Example of consumer preferences that switches from being concave to being convex

(Asked by: username_0 on 2023-01-25)

### Question

> Is there an example of consumer preferences over consumption bundles $(x,y)\in \Bbb R^2$ that would be ***concave*** when $x$ is abundant relative to $y$ and ***convex*** otherwise?

Are there known situations when this happens? I have never heard of that.

---

### Context

I've been recently thinking a lot about analyzing functions $f:\Bbb R \to \Bbb R$ with [positive third-order derivative](https://math.stackexchange.com/questions/4619836/why-are-monotonicity-and-convexity-so-intensely-studied-in-comparison-to-their-h) instead of functions with positive second-order derivative (convex functions). Such functions are either entirely convex / concave, or there is a unique point (inflection point) at which concavity switches into convexity. It turns out that such functions have many nice properties, for example they have at most three roots and most one local maximum and one local minimum. Therefore I expect that if a utility function $u(x,y)$ had positive/negative third-order derivative along lines in the space of bundles, then it would still be easy to analyze the consumer choice problem as it is in the case of convex preferences.

**Tags:** consumer-theory, preferences, choice-theory
> **Comments:**
> **username_1**: What are concave preferences? Convexity of preferences and convexity of functions are very different things.  
> **username_0**: @MichaelGreinecker I would regard preferences to be concave iff any lower contour set is concave. If the preference is represented by a utility function $u$, then $u$ is quasi-concave. I understand that concavity of $u$ is sufficient condition for concavity of the preferences, but not necessary. Note that function property of having positive third-order derivative can be also generalized to its "quasi" (i.e. ordinal) form.  
> **username_1**: Quasi-concavity of a representing utility function is equivalent to the convexity (!) of the represented preferences. I don’t know what concave sets are either.  
> **username_0**: @MichaelGreinecker Thanks a lot for the correction! So it is reversed: convex preferences $\Leftrightarrow$ quasi-concave utility; and concave preferences $\Leftrightarrow$ quasi-convex utility.  
> **username_1**: I'm still not sure what concave preferences should be? Convex upper contour sets?  
> **username_0**: I just see that I wrote a completely nonsense and I apologize for that: "preferences concave iff any lower contour set is concave" I meant that the set is convex. (A set can not be concave, at least that I know:-).) And I think that after this correction it is correct. In other terms, if I'm lees happy with either bundle $a$ or $b$ than with the bundle $c$, then I'm also less happy with any convex combination of $a$ and $b$ than with bundle $c$. Does this make sense? In simple terms: Convex preferences means preferring mixtures, concave preferences means disliking mixtures. Or am I confused?  
> **username_1**: You can define things like that. Convexity of preferences is a widely use assumption. I have never seen concavity of preferences.  
> **username_2**: @MichaelGreinecker I remember Varian made a hypothetical example of sardines and ice cream. Each one is delicious however combining them makes people worse off..  
> **username_1**: @T123 There are many reasons why non-convex preferences matter. It is not clear that this specific form is particularly useful.  
> **username_3**: The question would eventually have a meaning if you replace "consumer preferences" by "consumers' utility function"  
> **username_0**: @Bertrand Why won't the question make sense at it is? Just for a reference, convex and concave preferences are defined here: https://en.wikipedia.org/wiki/Preference_(economics)  
> **username_3**: @Pavel Kocourek. OK. I just wanted to be sure that there was no confusion (because you mentioned positive/negative third-order derivative of the utility function, which is unrelated to nonconvex preferences).  

---

## 1 answers

### Answer by username_4 on 2023-03-04. Score: 1

Consider the following form of utility function:

$U(x,y) = \beta (x-\alpha)^3 + y$, where $\alpha,\beta > 0$ are parameters. It is easy to check this utility function is monotone, in fact, strictly monotone.

For a constant utility level $\overline{U}$, we get the indifference curve:

$\overline{U} = \beta (x-\alpha)^3 + y \implies y = \overline{U} - \beta (x-\alpha)^3$.

Differentiating $y$ w.r.t. $x$ in the IC's:

$y' = - 3 \beta (x-\alpha)^2$

$y'' = -6 \beta (x-\alpha)$

We get that the IC's are convex for $x < \alpha$ and concave for $x > \alpha$.

Differentiating one more time,

$y''' = -6 \beta$.

So the IC's always have a negative third derivative.

Below is the indifference curve for the utility level $\overline{U} = 8$, for the functional form above, with parameters $\alpha = 2, \beta = 1$.

[![enter image description here](https://i.sstatic.net/8tG0S.png)](https://i.sstatic.net/8tG0S.png)


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