<p>As far as I can see this comes just from definitions:</p>
<p>As given in MWG definition 1.C.1:</p>
<blockquote>
<p>The choice structure <span class="math-container">$(\mathscr{B},C(\cdot))$</span> satisfies the weak axiom of revealed preference if the following property holds:</p>
</blockquote>
<blockquote>
<p>If for some <span class="math-container">$B \in \mathscr{B}$</span> with <span class="math-container">$x,y \in B$</span> we have <span class="math-container">$x\in C(B)$</span>, then for any <span class="math-container">$B'\in \mathscr{B}$</span> with <span class="math-container">$x,y\in B'$</span> and <span class="math-container">$y\in C(B')$</span>, we must also have <span class="math-container">$x\in C(B')$</span>.</p>
</blockquote>
<p>This basically states that under the weak axiom of revealed preference (WARP) if there is any choice set which contains both <span class="math-container">$x$</span> and <span class="math-container">$y$</span> and <span class="math-container">$x$</span> is preferred then there can be no set containing both <span class="math-container">$x$</span> and <span class="math-container">$y$</span> where <span class="math-container">$y$</span> would be chosen over <span class="math-container">$x$</span>. Or we can say that &quot;if x is revealed at least as good as y, then y cannot be revealed preferred to x.&quot;</p>
<p>Next according to MWG definition 3.B.2:</p>
<blockquote>
<p>The preference relation <span class="math-container">$\succeq$</span> on <span class="math-container">$X$</span> is monotone if <span class="math-container">$x\in X$</span> and <span class="math-container">$y &gt;&gt; x$</span> implies <span class="math-container">$y \succ x$</span>. It is strongly monotone if <span class="math-container">$y \geq x$</span> and <span class="math-container">$y \neq x$</span> imply that <span class="math-container">$y\succ x$</span>.</p>
</blockquote>
<p>In addition, according to MWG definition 3.B.4:</p>
<blockquote>
<p>The preference relation <span class="math-container">$\succeq $</span> on <span class="math-container">$X$</span> is convex if for every <span class="math-container">$x\in X$</span>, the upper countour set <span class="math-container">$\{y \in X: y \succeq x\}$</span> is convex; that is, if <span class="math-container">$y \succeq x$</span> and <span class="math-container">$z \succeq x$</span>, then <span class="math-container">$\alpha y + (1-\alpha) z \succeq x$</span> for any <span class="math-container">$\alpha \in [0,1]$</span>.</p>
</blockquote>
<p>Lastly, utility maximization is often used to imply rationality (<a href="http://www.sciencedirect.com/topics/computer-science/utility-maximization" rel="nofollow noreferrer">Simon 2001</a>) and preference relation can be rational only according to MWG definition 1.B.1:</p>
<blockquote>
<p>The preference relation <span class="math-container">$\succeq$</span> is rational if it possesses the following two properties:</p>
<p>(i) Completeness: for all <span class="math-container">$x,y \in X$</span> we have that <span class="math-container">$x\succeq y$</span> or <span class="math-container">$y \succeq x$</span> or both.</p>
<p>(ii) Transitivity: For all <span class="math-container">$x,y,z \in X$</span> if <span class="math-container">$x \succeq y$</span> and <span class="math-container">$y \succeq z$</span> then <span class="math-container">$x \succeq z$</span>.</p>
</blockquote>
<p>Given the definitions above if they are satisfied then if <span class="math-container">$y$</span> is revealed to be at least as good as <span class="math-container">$x$</span>, <span class="math-container">$x$</span> cannot later be revealed as better.  So it follows from the definitions of those properties. For example, WARP would be violated if the following choices are made <span class="math-container">$C(\{x,y\}) = y$</span> and in <span class="math-container">$C(\{x,y,z\}) = {x}$</span>, by a person holding their preferences constant - such situation would clearly violate several of the above definitions.</p>

CC BY-SA 4.0
2020-11-11T17:35:04.760Z
very explanatory! really many  thanks!
studentp
CC BY-SA 4.0
2020-11-10T23:05:58.953Z
2020-11-10T23:17:49.507Z
2020-11-10T23:17:49.507Z
1muflon1
<p>I read the following statement.</p>
<p>“ A utility maximizer with strictly convex and strongly monotonic preferences
satisfies weak axiom of revealed preferences.”</p>
<p>How can I prove or show this? I cannot realize this statement on my mind. Please show me it’s proof in order to understand its statement.</p>

CC BY-SA 4.0
2020-11-10T22:50:20.457Z
In my own opinion - it is must more instructive to think of WARP in terms of transitivity (consistency) of preferences under constrained decision-making. Deep down, WARP is about consistency, and more to the point: if consistency cannot be assumed in economic decision-making then it is hard to arrive at welfare conclusions from any impact on available choices. Any glance at an advanced micro text will give you the proof, but understanding why it is important is more informative.
EB3112
CC BY-SA 4.0
2020-11-10T22:55:00.083Z
Hmm thanks for comment. Which book? I look at MGW. But I cannot see @EB3112
studentp
CC BY-SA 4.0
2020-11-10T23:02:24.817Z
Tried Varian?...
EB3112
<microeconomics><utility><consumer-theory><preferences><self-study>
Proof on weak axiom of revealed preferences
studentp
# Proof on weak axiom of revealed preferences

(Asked by: username_0 on 2020-11-10)

I read the following statement.

“ A utility maximizer with strictly convex and strongly monotonic preferences
satisfies weak axiom of revealed preferences.”

How can I prove or show this? I cannot realize this statement on my mind. Please show me it’s proof in order to understand its statement.

**Tags:** microeconomics, utility, consumer-theory, preferences, self-study

> **Comments:**
> **username_1**: In my own opinion - it is must more instructive to think of WARP in terms of transitivity (consistency) of preferences under constrained decision-making. Deep down, WARP is about consistency, and more to the point: if consistency cannot be assumed in economic decision-making then it is hard to arrive at welfare conclusions from any impact on available choices. Any glance at an advanced micro text will give you the proof, but understanding why it is important is more informative.  
> **username_0**: Hmm thanks for comment. Which book? I look at MGW. But I cannot see @username_1  
> **username_1**: Tried Varian?...  

---

## 1 answers

### Answer by username_2 on 2020-11-10. Score: 4 | Accepted answer

As far as I can see this comes just from definitions:

As given in MWG definition 1.C.1:

> The choice structure $(\mathscr{B},C(\cdot))$ satisfies the weak axiom of revealed preference if the following property holds:

> If for some $B \in \mathscr{B}$ with $x,y \in B$ we have $x\in C(B)$, then for any $B'\in \mathscr{B}$ with $x,y\in B'$ and $y\in C(B')$, we must also have $x\in C(B')$.

This basically states that under the weak axiom of revealed preference (WARP) if there is any choice set which contains both $x$ and $y$ and $x$ is preferred then there can be no set containing both $x$ and $y$ where $y$ would be chosen over $x$. Or we can say that "if x is revealed at least as good as y, then y cannot be revealed preferred to x."

Next according to MWG definition 3.B.2:

> The preference relation $\succeq$ on $X$ is monotone if $x\in X$ and $y >> x$ implies $y \succ x$. It is strongly monotone if $y \geq x$ and $y \neq x$ imply that $y\succ x$.

In addition, according to MWG definition 3.B.4:

> The preference relation $\succeq $ on $X$ is convex if for every $x\in X$, the upper countour set $\{y \in X: y \succeq x\}$ is convex; that is, if $y \succeq x$ and $z \succeq x$, then $\alpha y + (1-\alpha) z \succeq x$ for any $\alpha \in [0,1]$.

Lastly, utility maximization is often used to imply rationality ([Simon 2001](http://www.sciencedirect.com/topics/computer-science/utility-maximization)) and preference relation can be rational only according to MWG definition 1.B.1:

> The preference relation $\succeq$ is rational if it possesses the following two properties:
>
> (i) Completeness: for all $x,y \in X$ we have that $x\succeq y$ or $y \succeq x$ or both.
>
> (ii) Transitivity: For all $x,y,z \in X$ if $x \succeq y$ and $y \succeq z$ then $x \succeq z$.

Given the definitions above if they are satisfied then if $y$ is revealed to be at least as good as $x$, $x$ cannot later be revealed as better. So it follows from the definitions of those properties. For example, WARP would be violated if the following choices are made $C(\{x,y\}) = y$ and in $C(\{x,y,z\}) = {x}$, by a person holding their preferences constant - such situation would clearly violate several of the above definitions.

> **Comments:**
> **username_0**: very explanatory! really many  thanks!  

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