<p>Every concave (convex) function is quasiconcave (quasiconvex).</p>
<p>Any nondecreasing transformation of a quasiconcave function is quasiconcave (i.e. if the function <span class="math-container">$f$</span> is quasiconcave and <span class="math-container">$g$</span> is a nondecreasing function, then <span class="math-container">$g\circ f$</span> is quasiconcave). This also means any nondecreasing transformation of a <em>concave</em> function is quasiconcave. (However it is <em>not</em> the case that every quasiconcave function is a nondecreasing transformation of some concave function.)</p>
<p>The same applies when &quot;concave&quot; replaced everywhere by &quot;convex&quot;.</p>
<hr />
<p>In economics, a preference relation <span class="math-container">$\succsim$</span> on the consumption set <span class="math-container">$X$</span> is called convex if, for all <span class="math-container">$x,y\in X$</span> with <span class="math-container">$y\succsim x$</span>, it is the case that for all <span class="math-container">$\lambda\in[0,1]$</span> that
<span class="math-container">$$\theta y +(1-\theta)x\succsim x$$</span></p>
<p>If the preference relation <span class="math-container">$\succsim$</span> can be represented by a utility function <span class="math-container">$u$</span> then the above condition can be written as: for  all <span class="math-container">$x,y\in X$</span> with <span class="math-container">$u(y)\geq u(x)$</span>, it is the case that for all <span class="math-container">$\lambda\in[0,1]$</span> that
<span class="math-container">$$u(\theta y +(1-\theta)x)\geq u(x).$$</span></p>
<p>But this is just the definition of quasiconcavity of <span class="math-container">$u$</span>. Thus if a utility function represents convex preferences it must be quasiconcave (and conversely).</p>
<hr />
<p>Take care not to confuse convexity of preferences with convexity of the utility function.</p>
<ul>
<li>As explained above, convexity of preferences is equivalent to quasi<em>concavity</em> of the utility function.</li>
<li>If the utility function is convex, then the utility function is quasiconvex, and quasiconvex functions are not generally quasiconcave. (A utility function that is both quasiconcave and quasiconvex must have indifference curves that are convex sets. In two dimensions, this corresponds to a straight line indifference curve.)</li>
</ul>

CC BY-SA 4.0
2023-08-31T03:46:56.570Z
It means the characteristics of preferences may be not the same with the utility function?
Huy Lê Thanh
CC BY-SA 4.0
2023-08-31T04:01:22.063Z
Example: the characteristic of preferences is convex but their utility function maybe convex or quasiconcave function, right?
Huy Lê Thanh
CC BY-SA 4.0
2023-08-31T09:56:10.120Z
If the preferences are convex, then the utility function is quasiconcave. If the utility function is quasiconcave, then the preferences are convex. If the utility function is convex, then it is quasiconvex, and preferences will not be convex unless the utility function happens to be quasiconcave as well. For example, the utility function $u(x,y)=x+y$ is concave and convex and so quasiconcave and quasiconvex. This utility function represents convex (but not strictly convex) preferences.
smcc
CC BY-SA 4.0
2023-08-30T15:15:39.260Z
2023-08-30T15:15:39.260Z
smcc
<p>I saw that the model of the quasi-convex utility function is similar to the concave utility function and also the quasi-concave utility function is similar to the convex utility function. How can distinguish them?</p>

CC BY-SA 4.0
2023-08-30T08:01:33.953Z
You can watch this playlist to better understand the terminology: https://youtube.com/playlist?list=PLUJGfL_499TLYEd-9IO1DmQKwm6Mtdt4u&feature=shared
Amit
<microeconomics><econometrics><mathematical-economics><utility><preferences>
Quasiconvex and quasiconcave utility function
Huy Lê Thanh
# Quasiconvex and quasiconcave utility function

(Asked by: username_0 on 2023-08-30)

I saw that the model of the quasi-convex utility function is similar to the concave utility function and also the quasi-concave utility function is similar to the convex utility function. How can distinguish them?

**Tags:** microeconomics, econometrics, mathematical-economics, utility, preferences
> **Comments:**
> **username_1**: You can watch this playlist to better understand the terminology: https://youtube.com/playlist?list=PLUJGfL_499TLYEd-9IO1DmQKwm6Mtdt4u&feature=shared  

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## 1 answers

### Answer by username_2 on 2023-08-30. Score: 2

Every concave (convex) function is quasiconcave (quasiconvex).

Any nondecreasing transformation of a quasiconcave function is quasiconcave (i.e. if the function $f$ is quasiconcave and $g$ is a nondecreasing function, then $g\circ f$ is quasiconcave). This also means any nondecreasing transformation of a *concave* function is quasiconcave. (However it is *not* the case that every quasiconcave function is a nondecreasing transformation of some concave function.)

The same applies when "concave" replaced everywhere by "convex".

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In economics, a preference relation $\succsim$ on the consumption set $X$ is called convex if, for all $x,y\in X$ with $y\succsim x$, it is the case that for all $\lambda\in[0,1]$ that
$$\theta y +(1-\theta)x\succsim x$$

If the preference relation $\succsim$ can be represented by a utility function $u$ then the above condition can be written as: for all $x,y\in X$ with $u(y)\geq u(x)$, it is the case that for all $\lambda\in[0,1]$ that
$$u(\theta y +(1-\theta)x)\geq u(x).$$

But this is just the definition of quasiconcavity of $u$. Thus if a utility function represents convex preferences it must be quasiconcave (and conversely).

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Take care not to confuse convexity of preferences with convexity of the utility function.

* As explained above, convexity of preferences is equivalent to quasi*concavity* of the utility function.
* If the utility function is convex, then the utility function is quasiconvex, and quasiconvex functions are not generally quasiconcave. (A utility function that is both quasiconcave and quasiconvex must have indifference curves that are convex sets. In two dimensions, this corresponds to a straight line indifference curve.)

> **Comments:**
> **username_0**: It means the characteristics of preferences may be not the same with the utility function?  
> **username_0**: Example: the characteristic of preferences is convex but their utility function maybe convex or quasiconcave function, right?  
> **username_2**: If the preferences are convex, then the utility function is quasiconcave. If the utility function is quasiconcave, then the preferences are convex. If the utility function is convex, then it is quasiconvex, and preferences will not be convex unless the utility function happens to be quasiconcave as well. For example, the utility function $u(x,y)=x+y$ is concave and convex and so quasiconcave and quasiconvex. This utility function represents convex (but not strictly convex) preferences.  

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