<blockquote>
  <p>Because WARP would imply that Barbara is chosen everywhere when Barbara is a choice in the set.</p>
</blockquote>

<p>This is not true. Instead WARP would say that <strong>it is not possible for Andrew to be chosen whenever both Andrew and Barbara are in the choice set</strong>, which is different than your claim above.</p>

<p>Just because $b$ is chosen from the set $\{a,b\}$ doesn't mean that $b$ should <em>always</em> be chosen, e.g. from the set $\{b,c\}$. It just means that $b$ is revealed-preferred to $a$, but it doesn't mean that $b$ is preferred to all other possible alternatives (in particular, $c$). For example, the preference $c\succ b\succ a$ is perfectly consistent with the revealed choice pattern of the firm.</p>

<p>Write out the two conditions explicitly: 
$$
\begin{align}
C(\color{red}{\varnothing},\color{red}{\{a\}},\{b\},\color{red}{\{a,b\}})&amp;=\{b\}\tag{1}\\
C(\color{red}{\varnothing},\color{red}{\{b\}},\{c\},\color{red}{\{b,c\}})&amp;=\{c\}\tag{2}
\end{align}
$$
$(1)$ just says that whenever the four choices --- hire no one, hire $a$ only, hire $b$ only, hire both --- are present, the firm hires $b$ only. 
Likewise, $(2)$ says whenever the four choices --- hire no one, hire $b$ only, hire $c$ only, hire both --- are present, the firm hires $c$ only. 
These two are consistent with WARP (for $(1)$, the x in your quoted definition is $a$ and y is $b$; for $(2)$, x is $b$ and y is $c$). </p>

<p>For restrictions on $C(2^{\{a,b,c\}})$, writing it out explicitly, we get 
$$
C(\color{red}{\varnothing},\color{red}{\{a\}},\color{red}{\{b\}},\color{black}{\{c\}},\color{red}{\{a,b\}},\color{black}{\{a,c\}},\color{red}{\{b,c\}},\color{black}{\{a,b,c\}})=\{\{a\},\{a,c\},\{a,b,c\}\}
$$
where the elements colored in red are revealed to be inferior according to $(1)$ and $(2)$.</p>

CC BY-SA 3.0
2017-09-29T22:22:43.867Z
2017-09-29T22:33:07.877Z
2017-09-29T22:33:07.877Z
Herr K.
<p>The following is a problem I am dealing with related to Weak Axiom of Revealed Preference. I have given my solution below to the situation. What I am not getting is how is WARP not violated?</p>

<p>A law firm looking to hire to fill three positions gets applications from Andrew, Barbara and Celia.</p>

<p>The law firm's set of alternatives is the set of possible hiring decisions:</p>

<p>$ X = \{ \phi, \{a\}, \{b\}, \{c\}, \{a,b\}, \{b,c\}, \{a,c\}, \{a,b,c\}  \} $</p>

<p>For any $Y \subset \{a,b,c\}$, define the power set of Y as</p>

<p>$ 2^{Y} \equiv \{Z | Z \subset Y \}$.</p>

<p>$ 2^{Y}$ is the set of hiring decisions that the firm can make when it receives applications from the lawyers in Y.</p>

<p>The law firm's budget sets $ B \in \mathcal{B} $ are the sets of hiring decisions it can make after receiving applications from some combination of Andrew, Barbara and Celia :</p>

<p>$\mathcal{B} = \{2^{Y} |Y \subset \{a, b, c \} \}$</p>

<p>1) When it receives applications from Andrew and Barbara, it will choose to hire Barbara (and not Andrew):-</p>

<p>$ C(2^{\{a,b\}}) = \{b\} $</p>

<p>2) When it receives applications from Barbara and Celia, it will choose to hire Celia (and not Barbara):</p>

<p>$ C(2^{\{b,c\}}) = \{c\} $</p>

<p>The following is the problem I am confused in :
Q) What restrictions does the weak axiom place on the firm's hiring
decision $C(2^{\{a,b,c\}})$ when it receives applications from Andrew, Barbara and Celia?</p>

<p>My Solution: </p>

<p>According to Mas-Colell et al (Definition 1.C.1) , the Weak Axiom of Revealed Preference says that if x is ever chosen when y is available then there can be no budget set containing both alternatives for which y is chosen and x is not.</p>

<p>So based on my understanding of WARP, in my situation above, when Andrew and Barbara apply the firm chooses Barbara , i.e. 
$ Barbara \succsim_R Andrew$
and when Barbara and Celia apply, the firm chooses Celia i.e. :
$ Celia \succsim_R Barbara $</p>

<p>Here we see that since Barbara is not chosen over Celia , WARP is violated. Because WARP would imply that Barbara is chosen everywhere when Barbara is a choice in the set. So when Andrew , Barbara and Celia apply, and WARP violates the above relation given , the firm would hire only Andrew.</p>

<p>What I am not getting is how is WARP not violated?</p>

<microeconomics><preferences>
Weak axiom of Revealed Preference application
Sky
# Weak axiom of Revealed Preference application

(Asked by: username_0 on 2017-09-29)

The following is a problem I am dealing with related to Weak Axiom of Revealed Preference. I have given my solution below to the situation. What I am not getting is how is WARP not violated?

A law firm looking to hire to fill three positions gets applications from Andrew, Barbara and Celia.

The law firm's set of alternatives is the set of possible hiring decisions:

$ X = \{ \phi, \{a\}, \{b\}, \{c\}, \{a,b\}, \{b,c\}, \{a,c\}, \{a,b,c\} \} $

For any $Y \subset \{a,b,c\}$, define the power set of Y as

$ 2^{Y} \equiv \{Z | Z \subset Y \}$.

$ 2^{Y}$ is the set of hiring decisions that the firm can make when it receives applications from the lawyers in Y.

The law firm's budget sets $ B \in \mathcal{B} $ are the sets of hiring decisions it can make after receiving applications from some combination of Andrew, Barbara and Celia :

$\mathcal{B} = \{2^{Y} |Y \subset \{a, b, c \} \}$

1) When it receives applications from Andrew and Barbara, it will choose to hire Barbara (and not Andrew):-

$ C(2^{\{a,b\}}) = \{b\} $

2) When it receives applications from Barbara and Celia, it will choose to hire Celia (and not Barbara):

$ C(2^{\{b,c\}}) = \{c\} $

The following is the problem I am confused in :
Q) What restrictions does the weak axiom place on the firm's hiring
decision $C(2^{\{a,b,c\}})$ when it receives applications from Andrew, Barbara and Celia?

My Solution:

According to Mas-Colell et al (Definition 1.C.1) , the Weak Axiom of Revealed Preference says that if x is ever chosen when y is available then there can be no budget set containing both alternatives for which y is chosen and x is not.

So based on my understanding of WARP, in my situation above, when Andrew and Barbara apply the firm chooses Barbara , i.e.
$ Barbara \succsim\_R Andrew$
and when Barbara and Celia apply, the firm chooses Celia i.e. :
$ Celia \succsim\_R Barbara $

Here we see that since Barbara is not chosen over Celia , WARP is violated. Because WARP would imply that Barbara is chosen everywhere when Barbara is a choice in the set. So when Andrew , Barbara and Celia apply, and WARP violates the above relation given , the firm would hire only Andrew.

What I am not getting is how is WARP not violated?

**Tags:** microeconomics, preferences

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## 1 answers

### Answer by username_1 on 2017-09-29. Score: 3 | Accepted answer

> Because WARP would imply that Barbara is chosen everywhere when Barbara is a choice in the set.

This is not true. Instead WARP would say that **it is not possible for Andrew to be chosen whenever both Andrew and Barbara are in the choice set**, which is different than your claim above.

Just because $b$ is chosen from the set $\{a,b\}$ doesn't mean that $b$ should *always* be chosen, e.g. from the set $\{b,c\}$. It just means that $b$ is revealed-preferred to $a$, but it doesn't mean that $b$ is preferred to all other possible alternatives (in particular, $c$). For example, the preference $c\succ b\succ a$ is perfectly consistent with the revealed choice pattern of the firm.

Write out the two conditions explicitly:
$$
\begin{align}
C(\color{red}{\varnothing},\color{red}{\{a\}},\{b\},\color{red}{\{a,b\}})&=\{b\}\tag{1}\\
C(\color{red}{\varnothing},\color{red}{\{b\}},\{c\},\color{red}{\{b,c\}})&=\{c\}\tag{2}
\end{align}
$$
$(1)$ just says that whenever the four choices --- hire no one, hire $a$ only, hire $b$ only, hire both --- are present, the firm hires $b$ only.
Likewise, $(2)$ says whenever the four choices --- hire no one, hire $b$ only, hire $c$ only, hire both --- are present, the firm hires $c$ only.
These two are consistent with WARP (for $(1)$, the x in your quoted definition is $a$ and y is $b$; for $(2)$, x is $b$ and y is $c$).

For restrictions on $C(2^{\{a,b,c\}})$, writing it out explicitly, we get
$$
C(\color{red}{\varnothing},\color{red}{\{a\}},\color{red}{\{b\}},\color{black}{\{c\}},\color{red}{\{a,b\}},\color{black}{\{a,c\}},\color{red}{\{b,c\}},\color{black}{\{a,b,c\}})=\{\{a\},\{a,c\},\{a,b,c\}\}
$$
where the elements colored in red are revealed to be inferior according to $(1)$ and $(2)$.


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