<p><strong>Q1</strong>:</p>
<blockquote>
<p>What are the differences between convex preferences and covex utility function</p>
</blockquote>
<p>They have different definitions, which imply different things. From <a href="https://en.wikipedia.org/wiki/Convex_preferences#Definition" rel="nofollow noreferrer">Wikipedia</a></p>
<blockquote>
<p>Formally, a preference relation <span class="math-container">$\succeq$</span> on the consumption set <span class="math-container">$X$</span> is called convex if whenever
<span class="math-container">$x, y, z \in X$</span> where <span class="math-container">$y \succeq x $</span> and <span class="math-container">$z \succeq x $</span>, then for every <span class="math-container">$\theta\in[0,1]$</span>: <span class="math-container">$$\theta y + (1-\theta) z \succeq x. $$</span></p>
</blockquote>
<p>while a function <span class="math-container">$f$</span> is convex, if, again from <a href="https://en.wikipedia.org/wiki/Convex_function#Definition" rel="nofollow noreferrer">Wikipedia</a>,</p>
<blockquote>
<p>For all <span class="math-container">$0 \leq t \leq 1$</span> and all <span class="math-container">$x_1, x_2 \in X$</span>:
<span class="math-container">$$f\left(t x_1 + (1-t) x_2\right) \leq t f\left(x_1\right) + (1-t) f\left(x_2\right)$$</span></p>
</blockquote>
<hr />
<p><strong>Q2</strong>:</p>
<blockquote>
<p>Why are convexity preferences usually represented by the quasi-concave function and not the convex function?</p>
</blockquote>
<p>The word convex appears in both, but that does not make it the same property, the definitions are different.</p>
<p>Just looking at the preference relation
<span class="math-container">$$\theta y + (1-\theta) z \succeq x $$</span>
from before, we get the utility equation
<span class="math-container">$$U(\theta y + (1-\theta) z) \geq U(x), $$</span>
which is not very similar to the definition of a convex function.</p>
<p>An example where a convex utility function represents a non-convex preference relation:</p>
<p>The utility function <span class="math-container">$U(x,y) = x^2 + y^2$</span> is convex (validate it using the definition).
Then <span class="math-container">$U(6,0) = U(0,6) = 36$</span>, while <span class="math-container">$U(4,4) = 32$</span> and <span class="math-container">$U(3,3) = 18$</span>. <span class="math-container">$(6,0)$</span> and <span class="math-container">$(0,6)$</span> are preferred to <span class="math-container">$(4,4)$</span>, as they have a higher utility.
The point <span class="math-container">$(3,3)$</span> is a convex combination of <span class="math-container">$(6,0)$</span> and <span class="math-container">$(0,6)$</span>. If the preference represented by this utility function was convex, according to the definition <span class="math-container">$(3,3)$</span> would also be preferred to <span class="math-container">$(4,4)$</span>, but it is not, it has a lower utility value.</p>

CC BY-SA 4.0
2024-02-17T20:04:43.863Z
2024-02-17T20:15:11.227Z
2024-02-17T20:15:11.227Z
Giskard
<p>What are the differences between convex preferences and covex utility function? Why are convexity preferences usually represented by the quasi-concave function and not the convex function?</p>

<microeconomics><utility><preferences>
Convex preference and convex utility
Huy Lê Thanh
# Convex preference and convex utility

(Asked by: username_0 on 2024-02-17)

What are the differences between convex preferences and covex utility function? Why are convexity preferences usually represented by the quasi-concave function and not the convex function?

**Tags:** microeconomics, utility, preferences

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## 1 answers

### Answer by username_1 on 2024-02-17. Score: 1

**Q1**:

> What are the differences between convex preferences and covex utility function

They have different definitions, which imply different things. From [Wikipedia](https://en.wikipedia.org/wiki/Convex_preferences#Definition)

> Formally, a preference relation $\succeq$ on the consumption set $X$ is called convex if whenever
> $x, y, z \in X$ where $y \succeq x $ and $z \succeq x $, then for every $\theta\in[0,1]$: $$\theta y + (1-\theta) z \succeq x. $$

while a function $f$ is convex, if, again from [Wikipedia](https://en.wikipedia.org/wiki/Convex_function#Definition),

> For all $0 \leq t \leq 1$ and all $x\_1, x\_2 \in X$:
> $$f\left(t x\_1 + (1-t) x\_2\right) \leq t f\left(x\_1\right) + (1-t) f\left(x\_2\right)$$

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**Q2**:

> Why are convexity preferences usually represented by the quasi-concave function and not the convex function?

The word convex appears in both, but that does not make it the same property, the definitions are different.

Just looking at the preference relation
$$\theta y + (1-\theta) z \succeq x $$
from before, we get the utility equation
$$U(\theta y + (1-\theta) z) \geq U(x), $$
which is not very similar to the definition of a convex function.

An example where a convex utility function represents a non-convex preference relation:

The utility function $U(x,y) = x^2 + y^2$ is convex (validate it using the definition).
Then $U(6,0) = U(0,6) = 36$, while $U(4,4) = 32$ and $U(3,3) = 18$. $(6,0)$ and $(0,6)$ are preferred to $(4,4)$, as they have a higher utility.
The point $(3,3)$ is a convex combination of $(6,0)$ and $(0,6)$. If the preference represented by this utility function was convex, according to the definition $(3,3)$ would also be preferred to $(4,4)$, but it is not, it has a lower utility value.


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