Question on The Weak Axiom of Revealed Preference and The Definition of Revealed Preference Relation

Question

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. 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!

Answer 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?