<p>Let <span class="math-container">${\cal B}$</span> be a collection of choice sets and let <span class="math-container">$C$</span> be a single valued choice function, picking one item <span class="math-container">$C(B)$</span> out of every set <span class="math-container">$B \in {\cal B}$</span>.</p>
<p>Let <span class="math-container">$x$</span> be revealed preferred to <span class="math-container">$y$</span> if there is some <span class="math-container">$B \in {\cal B}$</span>, such that <span class="math-container">$x,y \in B$</span>, <span class="math-container">$x \in C(B)$</span> and <span class="math-container">$y \ne x$</span> (which implies <span class="math-container">$y \notin C(B)$</span>. We can denote this by <span class="math-container">$x \succeq_R y$</span>.</p>
<p>In the first part of our question you establish the conditions such that <span class="math-container">$x(p_1,w_1) \succeq_R x(p_2, w_2)$</span> and not <span class="math-container">$x(p_2, w_2) \succeq_R x(p_1, w_1)$</span>.</p>
<p>For the second part of your question, you only check the condition <span class="math-container">$x(p_1,w_1) \succeq_R x(p_2, w_2)$</span>.</p>
<p>Note that (in principle) the revealed preference relation does not need to be asymmetric. So the second part leaves open the possiblity that also <span class="math-container">$x(p_2, w_2) \succeq_R x(p_1, w_1)$</span>.</p>
<p>Notice that the question asks: Over what range of quantities of good 2 consumed in year 2 <strong>would you conclude</strong> that the consumer's consumption bundle in year 1 is revealed preferred to that in year 2?</p>
<p>It does not ask over what range of quantities of good 2 is the consumer's consumption bundle in year 1 revealed preferred (as by definition) to that in year 2?</p>

CC BY-SA 4.0
2024-02-09T20:38:06.837Z
Thanks a lot for your answer! However, I couldn't agree with you on that. Here is my thought, please correct me if I am wrong: In the second way, other than $x(\mathbf{p}^1,w_1)\succsim^* x(\mathbf{p}^2,w_2)$, I also checked that $x(\mathbf{p}^1,w_1)\in C(B_{\mathbf{p}^1,w_1})$ but $x(\mathbf{p}^2,w_2)\notin C(B_{\mathbf{p}^1,w_1})$ (although it's trivial as C(B_{\mathbf{p}^1,w_1}) is a singleton containing only x(\mathbf{p}^1,w_1)). Indeed, when $75\leq y\leq80$, we'd also have $x(\mathbf{p}^2,w_2)\succsim^*B_{\mathbf{p}^1,w_1}$. But it shouldn't be an issue,
Shenron
CC BY-SA 4.0
2024-02-09T20:38:44.543Z
(cont'd) because such a preference can only be "revealed" by the observed choice behavior $C(B_{\mathbf{p}^2,w_2})=\{x(\mathbf{p}^2,w_2)\}$ when $x(\mathbf{p}^2,w_1),x(\mathbf{p}^2,w_2)\in B_{\mathbf{p}^2,w_2}$. Moreover, in the textbook (page 11 right after Definition 1.C.2), it says that "the revealed preference relation $\succsim^*$ need not be either complete or transitive."
Shenron
CC BY-SA 4.0
2024-02-09T20:39:07.853Z
(cont'd) Therefore, I guess the point here might be that the question implicitly assumed that the consumer's behavior is consistent (that is, it satisfies the weak axiom). (After all, in part (a) of the exercise, it asked over what range would his behavior be inconsistent.) Really sorry for this long comment, please let me know your thought. I really appreciate it!
Shenron
CC BY-SA 4.0
2024-02-09T08:34:38.630Z
2024-02-09T08:34:38.630Z
tdm
<p>I am solving the following problem (from Exercise 2.F.3 (b) in MWG) and I got confused by the weak axiom of revealed preference and the definition of the revealed preference relation. Here is the exercise question:</p>
<h2>Question</h2>
<blockquote>
<p>You are given the following partial information about a consumer's purchases. He consumes only two goods.
<a href="https://i.sstatic.net/77YXr.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/77YXr.png" alt="enter image description here" /></a>
Over what range of quantities of good 2 consumed in year 2 would you conclude that the consumer's consumption bundle in year 1 is revealed preferred to that in year 2?</p>
</blockquote>
<h2>My Methods</h2>
<p>I have two pieces of thought about this exercise. The first one is based on the weak axiom of revealed preference, and it does work. However, the second one, which is based on the definition of the revealed preference relation, does not work. Let me elaborate it:</p>
<p>Denote <span class="math-container">$\mathbf{p}^1 = (p_1^1,p_2^1) = (100,100)$</span>, <span class="math-container">$\mathbf{p}^2 = (p_1^2,p_2^2) = (100,80)$</span>, <span class="math-container">$w_1 = 100 \times 100 + 100 \times 100 = 20000$</span>, <span class="math-container">$w_2 = 120 \times 100 + 80y = 12000 + 80y$</span>, <span class="math-container">$x(\mathbf{p}^1,w_1) = \{(100,100)\}$</span>, and <span class="math-container">$x(\mathbf{p}^2,w_2) = \{(120,y)\}$</span>, where <span class="math-container">$y$</span> is the quantity of consumption of good 2 in year 2.</p>
<p><em><strong>As for my first method</strong></em>, in the textbook (MWG, page 29), it says the following:</p>
<blockquote>
<p>In the consumer demand setting, the idea behind the weak axiom can be put as follows: If <span class="math-container">$\mathbf{p} \cdot x(\mathbf{p}',w') \leq w$</span> and <span class="math-container">$x(\mathbf{p},w) \neq x(\mathbf{p}',w')$</span>, then we know that when facing prices <span class="math-container">$\mathbf{p}$</span> and wealth <span class="math-container">$w$</span>, the consumer chose consumption bundle <span class="math-container">$x(\mathbf{p},w)$</span> even though bundle <span class="math-container">$x(\mathbf{p}',w')$</span> was also affordable, We can interpret this choice as &quot;revealing&quot; a preference for <span class="math-container">$x(\mathbf{p},w)$</span> over <span class="math-container">$x(\mathbf{p}',w')$</span>. Now, we might reasonably expect the consumer to display some consistency in his demand behavior. In particular, given his revealed preference, we expect that he would choose <span class="math-container">$x(\mathbf{p},w)$</span> over <span class="math-container">$x(\mathbf{p}',w')$</span> whenever they are both affordable, If so, bundle <span class="math-container">$x(\mathbf{p},w)$</span> must not be affordable at the price-wealth combination <span class="math-container">$(\mathbf{p}',w')$</span> at which the consumer chooses bundle <span class="math-container">$x(\mathbf{p}',w')$</span>. THat is, as required by the weak axiom, we must have <span class="math-container">$\mathbf{p}' \cdot x(\mathbf{p},w) &gt; w'$</span>.</p>
</blockquote>
<p>Hence, we must have that, when facing the price-wealth combination <span class="math-container">$(\mathbf{p}^1,w_1)$</span>, both <span class="math-container">$x(\mathbf{p}^1,w_1)$</span> and <span class="math-container">$x(\mathbf{p}^2,w_2)$</span> are affordable, and that <span class="math-container">$x(\mathbf{p}^1,w_1)$</span> must not be affordable at the price-wealth combination <span class="math-container">$(\mathbf{p}^2,w_2)$</span>. Therefore,
<span class="math-container">\begin{align*}
&amp;\mathbf{p}^1 \cdot x(\mathbf{p}^2,w_2) = 100 \times 120 + 100y \leq 20000,\ \text{and}\\
&amp;\mathbf{p}^2 \cdot x(\mathbf{p}^1,w_1) = 100 \times 100 + 80 \times 100 &gt; 12000 + 80y.
\end{align*}</span>
These inequalities gives us <span class="math-container">$y &lt; 75$</span>, which is the correct answer.</p>
<p><em><strong>My second method</strong></em>, on the other hand, does not involve the weak axiom, but instead, apply directly the definition of the revealed preference relation (Definition 1.C.2):</p>
<blockquote>
<p><strong>Definition 1.C.2</strong><span class="math-container">$\quad$</span> Given a choice structure <span class="math-container">$(\mathcal{B},C(\cdot))$</span>, the <em>revealed preference relation</em> <span class="math-container">$\succsim^*$</span> is defined by
<span class="math-container">\begin{align*}
x \succsim^* y \iff \text{there is some $B \in \mathcal{B}$ such that $x, y \in B$ and $x \in C(B)$}.
\end{align*}</span>
Moreover, we say that &quot;<span class="math-container">$x$</span> is revealed preferred to <span class="math-container">$y$</span>&quot; if there is some <span class="math-container">$B \in \mathcal{B}$</span> such that <span class="math-container">$x, y \in B$</span>, <span class="math-container">$x \in C(B)$</span>, and <span class="math-container">$y \notin C(B)$</span>.</p>
</blockquote>
<p>Based on this definition, I tried to solve this exercise as follows: Suppose that <span class="math-container">$x(\mathbf{p}^1,w_1), x(\mathbf{p}^2,w_2) \in B_{\mathbf{p}^1,w_1} = \left\{z \in \mathbb{R}_+^2: \mathbf{p}^1 \cdot z \leq w_1\right\}$</span>. Then, <span class="math-container">$\mathbf{p}^1 \cdot x(\mathbf{p}^1,w_1) \leq w_1$</span>, which is always true; and <span class="math-container">$\mathbf{p}^1 \cdot x(\mathbf{p}^2,w_2) = 100 \times 120 + 100y \leq 20000$</span>, which implies that <span class="math-container">$y \leq 80$</span>. For <span class="math-container">$x(\mathbf{P}^1,w_1)$</span> to be revealed preferred to <span class="math-container">$x(\mathbf{p}^2,w_2)$</span>, we need <span class="math-container">$x(\mathbf{P}^1,w_1) \in C(B_{\mathbf{p}^1,w_1})$</span> but <span class="math-container">$x(\mathbf{P}^2,w_2) \notin C(B_{\mathbf{p}^1,w_1})$</span>. However, this condition is automatically satisfied, because in the context of Walrasian demand, <span class="math-container">$C(B_{\mathbf{p}^1,w_1}) = x(\mathbf{p}^1,w_1) = \{(100,100)\} \neq \{(120,y)\} = x(\mathbf{p}^2,w_2)$</span>, no matter what <span class="math-container">$y$</span> is. So, I concluded that <span class="math-container">$y \leq 80$</span> would make <span class="math-container">$x(\mathbf{p}^1,w_1)$</span> be revealed preferred to <span class="math-container">$x(\mathbf{p}^2,w_2)$</span>.</p>
<h2>My Questions</h2>
<p>Could someone please tell me where did I do wrong in the second method? What did I miss? I really appreciate it!</p>

<microeconomics><mathematical-economics><consumer-theory><preferences><decision-theory>
Question on The Weak Axiom of Revealed Preference and The Definition of Revealed Preference Relation
Shenron
# Question on The Weak Axiom of Revealed Preference and The Definition of Revealed Preference Relation

(Asked by: username_0 on 2024-02-08)

I am solving the following problem (from Exercise 2.F.3 (b) in MWG) and I got confused by the weak axiom of revealed preference and the definition of the revealed preference relation. Here is the exercise question:

## Question

> You are given the following partial information about a consumer's purchases. He consumes only two goods.
> [![enter image description here](https://i.sstatic.net/77YXr.png)](https://i.sstatic.net/77YXr.png)
> Over what range of quantities of good 2 consumed in year 2 would you conclude that the consumer's consumption bundle in year 1 is revealed preferred to that in year 2?

## My Methods

I have two pieces of thought about this exercise. The first one is based on the weak axiom of revealed preference, and it does work. However, the second one, which is based on the definition of the revealed preference relation, does not work. Let me elaborate it:

Denote $\mathbf{p}^1 = (p\_1^1,p\_2^1) = (100,100)$, $\mathbf{p}^2 = (p\_1^2,p\_2^2) = (100,80)$, $w\_1 = 100 \times 100 + 100 \times 100 = 20000$, $w\_2 = 120 \times 100 + 80y = 12000 + 80y$, $x(\mathbf{p}^1,w\_1) = \{(100,100)\}$, and $x(\mathbf{p}^2,w\_2) = \{(120,y)\}$, where $y$ is the quantity of consumption of good 2 in year 2.

***As for my first method***, in the textbook (MWG, page 29), it says the following:

> In the consumer demand setting, the idea behind the weak axiom can be put as follows: If $\mathbf{p} \cdot x(\mathbf{p}',w') \leq w$ and $x(\mathbf{p},w) \neq x(\mathbf{p}',w')$, then we know that when facing prices $\mathbf{p}$ and wealth $w$, the consumer chose consumption bundle $x(\mathbf{p},w)$ even though bundle $x(\mathbf{p}',w')$ was also affordable, We can interpret this choice as "revealing" a preference for $x(\mathbf{p},w)$ over $x(\mathbf{p}',w')$. Now, we might reasonably expect the consumer to display some consistency in his demand behavior. In particular, given his revealed preference, we expect that he would choose $x(\mathbf{p},w)$ over $x(\mathbf{p}',w')$ whenever they are both affordable, If so, bundle $x(\mathbf{p},w)$ must not be affordable at the price-wealth combination $(\mathbf{p}',w')$ at which the consumer chooses bundle $x(\mathbf{p}',w')$. THat is, as required by the weak axiom, we must have $\mathbf{p}' \cdot x(\mathbf{p},w) > w'$.

Hence, we must have that, when facing the price-wealth combination $(\mathbf{p}^1,w\_1)$, both $x(\mathbf{p}^1,w\_1)$ and $x(\mathbf{p}^2,w\_2)$ are affordable, and that $x(\mathbf{p}^1,w\_1)$ must not be affordable at the price-wealth combination $(\mathbf{p}^2,w\_2)$. Therefore,
\begin{align\*}
&\mathbf{p}^1 \cdot x(\mathbf{p}^2,w\_2) = 100 \times 120 + 100y \leq 20000,\ \text{and}\\
&\mathbf{p}^2 \cdot x(\mathbf{p}^1,w\_1) = 100 \times 100 + 80 \times 100 > 12000 + 80y.
\end{align\*}
These inequalities gives us $y < 75$, which is the correct answer.

***My second method***, on the other hand, does not involve the weak axiom, but instead, apply directly the definition of the revealed preference relation (Definition 1.C.2):

> **Definition 1.C.2**$\quad$ Given a choice structure $(\mathcal{B},C(\cdot))$, the *revealed preference relation* $\succsim^\*$ is defined by
> \begin{align\*}
> x \succsim^\* y \iff \text{there is some $B \in \mathcal{B}$ such that $x, y \in B$ and $x \in C(B)$}.
> \end{align\*}
> Moreover, we say that "$x$ is revealed preferred to $y$" if there is some $B \in \mathcal{B}$ such that $x, y \in B$, $x \in C(B)$, and $y \notin C(B)$.

Based on this definition, I tried to solve this exercise as follows: Suppose that $x(\mathbf{p}^1,w\_1), x(\mathbf{p}^2,w\_2) \in B\_{\mathbf{p}^1,w\_1} = \left\{z \in \mathbb{R}\_+^2: \mathbf{p}^1 \cdot z \leq w\_1\right\}$. Then, $\mathbf{p}^1 \cdot x(\mathbf{p}^1,w\_1) \leq w\_1$, which is always true; and $\mathbf{p}^1 \cdot x(\mathbf{p}^2,w\_2) = 100 \times 120 + 100y \leq 20000$, which implies that $y \leq 80$. For $x(\mathbf{P}^1,w\_1)$ to be revealed preferred to $x(\mathbf{p}^2,w\_2)$, we need $x(\mathbf{P}^1,w\_1) \in C(B\_{\mathbf{p}^1,w\_1})$ but $x(\mathbf{P}^2,w\_2) \notin C(B\_{\mathbf{p}^1,w\_1})$. However, this condition is automatically satisfied, because in the context of Walrasian demand, $C(B\_{\mathbf{p}^1,w\_1}) = x(\mathbf{p}^1,w\_1) = \{(100,100)\} \neq \{(120,y)\} = x(\mathbf{p}^2,w\_2)$, no matter what $y$ is. So, I concluded that $y \leq 80$ would make $x(\mathbf{p}^1,w\_1)$ be revealed preferred to $x(\mathbf{p}^2,w\_2)$.

## My Questions

Could someone please tell me where did I do wrong in the second method? What did I miss? I really appreciate it!

**Tags:** microeconomics, mathematical-economics, consumer-theory, preferences, decision-theory

---

## 1 answers

### Answer by username_1 on 2024-02-09. Score: 1

Let ${\cal B}$ be a collection of choice sets and let $C$ be a single valued choice function, picking one item $C(B)$ out of every set $B \in {\cal B}$.

Let $x$ be revealed preferred to $y$ if there is some $B \in {\cal B}$, such that $x,y \in B$, $x \in C(B)$ and $y \ne x$ (which implies $y \notin C(B)$. We can denote this by $x \succeq\_R y$.

In the first part of our question you establish the conditions such that $x(p\_1,w\_1) \succeq\_R x(p\_2, w\_2)$ and not $x(p\_2, w\_2) \succeq\_R x(p\_1, w\_1)$.

For the second part of your question, you only check the condition $x(p\_1,w\_1) \succeq\_R x(p\_2, w\_2)$.

Note that (in principle) the revealed preference relation does not need to be asymmetric. So the second part leaves open the possiblity that also $x(p\_2, w\_2) \succeq\_R x(p\_1, w\_1)$.

Notice that the question asks: Over what range of quantities of good 2 consumed in year 2 **would you conclude** that the consumer's consumption bundle in year 1 is revealed preferred to that in year 2?

It does not ask over what range of quantities of good 2 is the consumer's consumption bundle in year 1 revealed preferred (as by definition) to that in year 2?

> **Comments:**
> **username_0**: Thanks a lot for your answer! However, I couldn't agree with you on that. Here is my thought, please correct me if I am wrong: In the second way, other than $x(\mathbf{p}^1,w_1)\succsim^* x(\mathbf{p}^2,w_2)$, I also checked that $x(\mathbf{p}^1,w_1)\in C(B_{\mathbf{p}^1,w_1})$ but $x(\mathbf{p}^2,w_2)\notin C(B_{\mathbf{p}^1,w_1})$ (although it's trivial as C(B_{\mathbf{p}^1,w_1}) is a singleton containing only x(\mathbf{p}^1,w_1)). Indeed, when $75\leq y\leq80$, we'd also have $x(\mathbf{p}^2,w_2)\succsim^*B_{\mathbf{p}^1,w_1}$. But it shouldn't be an issue,  
> **username_0**: (cont'd) because such a preference can only be "revealed" by the observed choice behavior $C(B_{\mathbf{p}^2,w_2})=\{x(\mathbf{p}^2,w_2)\}$ when $x(\mathbf{p}^2,w_1),x(\mathbf{p}^2,w_2)\in B_{\mathbf{p}^2,w_2}$. Moreover, in the textbook (page 11 right after Definition 1.C.2), it says that "the revealed preference relation $\succsim^*$ need not be either complete or transitive."  
> **username_0**: (cont'd) Therefore, I guess the point here might be that the question implicitly assumed that the consumer's behavior is consistent (that is, it satisfies the weak axiom). (After all, in part (a) of the exercise, it asked over what range would his behavior be inconsistent.) Really sorry for this long comment, please let me know your thought. I really appreciate it!  

---

