<p>Let <span class="math-container">$X$</span> be the convex set of alternatives, let <span class="math-container">$\succeq$</span> be a preference relation and let <span class="math-container">$u(.)$</span> be a utility function that reflects these preferences, which means that <span class="math-container">$u(x) \ge u(y)$</span> if and only if <span class="math-container">$x \succeq y$</span>.</p>
<p>The <strong>preference relation <span class="math-container">$\succeq$</span> is convex</strong> if</p>
<ul>
<li><p>For all <span class="math-container">$y$</span>, <span class="math-container">$x$</span> and <span class="math-container">$z$</span> in <span class="math-container">$X$</span>, if <span class="math-container">$x \succeq y$</span> and <span class="math-container">$z \succeq y$</span> then for all <span class="math-container">$\alpha \in [0,1]$</span>, <span class="math-container">$\alpha x + (1-\alpha) y \succeq y$</span>.</p>
</li>
<li><p>Equivalently, for all <span class="math-container">$y$</span> in <span class="math-container">$X$</span>, the set of all bundles that are at least as good as <span class="math-container">$y$</span>, is a convex set.</p>
</li>
<li><p>Equivalently, for all <span class="math-container">$y$</span> in <span class="math-container">$X$</span>, the set <span class="math-container">$U_y = \{x \in X| x \succeq y\}$</span> is convex.</p>
</li>
</ul>
<p>The <strong>utility function</strong> <span class="math-container">$u$</span> is <strong>Quasi-concave</strong> if</p>
<ul>
<li><p>For all <span class="math-container">$x$</span>, <span class="math-container">$y$</span> and <span class="math-container">$z$</span> in <span class="math-container">$X$</span>, if <span class="math-container">$u(x) \ge u(y)$</span> and <span class="math-container">$u(z) \ge u(y)$</span>, then for all <span class="math-container">$\alpha \in [0,1]$</span>, <span class="math-container">$u(\alpha x + (1-\alpha) z)\ge u(y)$</span>.</p>
</li>
<li><p>Equivalently, for all <span class="math-container">$y$</span> in <span class="math-container">$X$</span>, the set of bundels that give at least as much utility as <span class="math-container">$y$</span> is a convex set.</p>
</li>
<li><p>Equivalently, for all <span class="math-container">$y \in X$</span> the set <span class="math-container">$V_y = \{x \in X| u(x) \ge u(y)\}$</span> is convex.</p>
</li>
</ul>
<p>However the set <span class="math-container">$U_y$</span> and <span class="math-container">$V_y$</span> are the same.
<span class="math-container">$$
 V_y = \{x \in X| u(x) \ge u(y)\} = \{x \in X| x \succeq y\} = U_y.
$$</span>
As such, convexity of preferences is identical to quasi-concavity of the utility function that reflects these preferences.</p>
<p>Intuitively, convexity of preferences means that all upper contour sets are convex sets. This is by definition equal to convexity of all sets that give at least as much utility level than some given fixed bundle. This is identicial to the notion of quasi-concavity.</p>

CC BY-SA 4.0
2022-01-08T16:32:59.293Z
Hi @tdm. Could you expand a little on the intuition at the end there, if possible?
EB3112
CC BY-SA 4.0
2022-01-09T14:07:29.023Z
Aren't the second and third points about convex preference relations the exact same thing? As in, they are not equivalent definitions, but the exact same statement with a different notation? Same goes for the second and third points about quasi-concave functions.
Giskard
CC BY-SA 4.0
2022-01-09T17:43:08.580Z
@Giskard of course, that is why I said "Equivalently". The second is in words, the third in more formal notation. I was just trying to be more pedagogical.
tdm
CC BY-SA 4.0
2022-01-09T18:40:40.070Z
I applaud your pedagogical intentions. You also wrote "Equivalently" between the first and second points, which are equivalent definitions. But the second and third points are not merely equivalent, but the exact same definition with different notation. Instead of "equivalently" I would prefer to use "or with more formal notation" or "i.e.". But I acknowledge that this is mostly a matter of taste.
Giskard
CC BY-SA 4.0
2022-01-08T09:55:59.737Z
2022-01-09T14:05:25.097Z
2022-01-09T14:05:25.097Z
tdm
<p>How does convex preference imply quasi concave utility function (Intuitively and Mathematically)?</p>

<microeconomics><consumer-theory>
Convex Preference and utility function
Arka Mukherjee
# Convex Preference and utility function

(Asked by: username_0 on 2022-01-08)

How does convex preference imply quasi concave utility function (Intuitively and Mathematically)?

**Tags:** microeconomics, consumer-theory

---

## 1 answers

### Answer by username_1 on 2022-01-08. Score: 7

Let $X$ be the convex set of alternatives, let $\succeq$ be a preference relation and let $u(.)$ be a utility function that reflects these preferences, which means that $u(x) \ge u(y)$ if and only if $x \succeq y$.

The **preference relation $\succeq$ is convex** if

* For all $y$, $x$ and $z$ in $X$, if $x \succeq y$ and $z \succeq y$ then for all $\alpha \in [0,1]$, $\alpha x + (1-\alpha) y \succeq y$.
* Equivalently, for all $y$ in $X$, the set of all bundles that are at least as good as $y$, is a convex set.
* Equivalently, for all $y$ in $X$, the set $U\_y = \{x \in X| x \succeq y\}$ is convex.

The **utility function** $u$ is **Quasi-concave** if

* For all $x$, $y$ and $z$ in $X$, if $u(x) \ge u(y)$ and $u(z) \ge u(y)$, then for all $\alpha \in [0,1]$, $u(\alpha x + (1-\alpha) z)\ge u(y)$.
* Equivalently, for all $y$ in $X$, the set of bundels that give at least as much utility as $y$ is a convex set.
* Equivalently, for all $y \in X$ the set $V\_y = \{x \in X| u(x) \ge u(y)\}$ is convex.

However the set $U\_y$ and $V\_y$ are the same.
$$
V\_y = \{x \in X| u(x) \ge u(y)\} = \{x \in X| x \succeq y\} = U\_y.
$$
As such, convexity of preferences is identical to quasi-concavity of the utility function that reflects these preferences.

Intuitively, convexity of preferences means that all upper contour sets are convex sets. This is by definition equal to convexity of all sets that give at least as much utility level than some given fixed bundle. This is identicial to the notion of quasi-concavity.

> **Comments:**
> **username_2**: Hi @tdm. Could you expand a little on the intuition at the end there, if possible?  
> **username_3**: Aren't the second and third points about convex preference relations the exact same thing? As in, they are not equivalent definitions, but the exact same statement with a different notation? Same goes for the second and third points about quasi-concave functions.  
> **username_1**: @Giskard of course, that is why I said "Equivalently". The second is in words, the third in more formal notation. I was just trying to be more pedagogical.  
> **username_3**: I applaud your pedagogical intentions. You also wrote "Equivalently" between the first and second points, which are equivalent definitions. But the second and third points are not merely equivalent, but the exact same definition with different notation. Instead of "equivalently" I would prefer to use "or with more formal notation" or "i.e.". But I acknowledge that this is mostly a matter of taste.  

---

