<p>Your definition of WARP needs correction.
The Walrasian demand function <span class="math-container">$x(p,w)$</span> satisfies the weak axiom of revealed preference if the following property holds:
<span class="math-container">$(\forall (p,w), (p',w'))(( p\cdot x(p',w')\leq w \ \wedge \ x(p',w')\neq x(p,w))\Rightarrow p'\cdot x(p,w)&gt;w')$</span></p>
<p>The negation of WARP can be written as the following property:
<span class="math-container">$(\exists (p,w), (p',w'))(( p\cdot x(p',w')\leq w \ \wedge \ x(p',w')\neq x(p,w))\wedge p'\cdot x(p,w)\leq w')$</span></p>

CC BY-SA 4.0
2024-10-04T14:09:39.593Z
Thanks, corrected. Could you help me with same definition but in context of choice structure:

If for some $B \in \mathcal{B}$ with $x,y \in B$ we have $x \in C(B)$, then for any $B' \in \mathcal{B}$ with $x,y \in B'$ and $y \in \ C(B')$, we must also have $x \in C(B')$
hr08
CC BY-SA 4.0
2024-10-04T14:22:53.193Z
Do you want its negation?
Amit
CC BY-SA 4.0
2024-10-04T14:34:54.657Z
yes  , because i want to know what conditions come under antecedent and for WARP to be vacuously the antecedent must include conditions upto ...$y \in C(B')$
hr08
CC BY-SA 4.0
2024-10-04T14:37:19.140Z
Here is the negation of WARP: There exist $B,B'\in\mathcal{B}$ and there exist $x,y\in B\cap B'$ such that $x\in C(B)$, $y\in C(B')$ but $x\notin C(B')$.
Amit
CC BY-SA 4.0
2024-10-04T14:55:01.163Z
Definition of WARP is $(\forall B, B'\in\mathcal{B})(\forall x,y\in B\cap B')((x\in C(B)\wedge y\in C(B'))\Rightarrow x\in C(B'))$
Amit
CC BY-SA 4.0
2024-10-04T14:55:51.410Z
Thanks alot @Amit Sir
hr08
CC BY-SA 4.0
2024-10-04T13:56:27.437Z
2024-10-04T13:56:27.437Z
Amit
<p>I wanted to write the negation of WARP.
The following is the definition of WARP from <em>Microeconomic Theory by Andreu Mas-Colell, Whinston and Green</em>:</p>
<p>The Walrasian demand function <span class="math-container">$x(p,w)$</span> satisfies the weak axiom of revealed preferences if the following property holds for any two price-wealth situations <span class="math-container">$(p,w)$</span> and <span class="math-container">$(p',w')$</span>:</p>
<p>If <span class="math-container">$p.x(p',w') \leq w$</span> and <span class="math-container">$x(p',w') \neq x(p,w)$</span> then <span class="math-container">$p'.x(p,w)&gt;w'$</span></p>
<p>My attempt:</p>
<p><span class="math-container">$\exists$</span> <span class="math-container">$(p,w)$</span>, <span class="math-container">$(p',w')$</span> : <span class="math-container">$(p.x(p',w') \leq w) \land ((x(p',w') \neq x(p,w))$</span> <span class="math-container">$\land$</span> <span class="math-container">$(p'.x(p,w)\leq w')$</span></p>
<p>Is this correct and how to write the negation statement in English language similar to how it is written in book.</p>

CC BY-SA 4.0
2024-10-04T13:59:27.347Z
The first inequality in the definition should have been $p\cdot x(p',w')\le w$.
Herr K.
CC BY-SA 4.0
2024-10-04T14:00:13.420Z
thanks corrected
hr08
CC BY-SA 4.0
2024-10-04T14:01:13.713Z
The first inequality in the negation should correspondingly be corrected. Then all look good.
Herr K.
CC BY-SA 4.0
2024-10-04T14:02:18.120Z
thanks @HerrK. , corrected
hr08
<microeconomics><consumer-theory>
Negation of WARP
hr08
# Negation of WARP

(Asked by: username_0 on 2024-10-04)

I wanted to write the negation of WARP.
The following is the definition of WARP from *Microeconomic Theory by Andreu Mas-Colell, Whinston and Green*:

The Walrasian demand function $x(p,w)$ satisfies the weak axiom of revealed preferences if the following property holds for any two price-wealth situations $(p,w)$ and $(p',w')$:

If $p.x(p',w') \leq w$ and $x(p',w') \neq x(p,w)$ then $p'.x(p,w)>w'$

My attempt:

$\exists$ $(p,w)$, $(p',w')$ : $(p.x(p',w') \leq w) \land ((x(p',w') \neq x(p,w))$ $\land$ $(p'.x(p,w)\leq w')$

Is this correct and how to write the negation statement in English language similar to how it is written in book.

**Tags:** microeconomics, consumer-theory
> **Comments:**
> **username_1**: The first inequality in the definition should have been $p\cdot x(p',w')\le w$.  
> **username_0**: thanks corrected  
> **username_1**: The first inequality in the negation should correspondingly be corrected. Then all look good.  
> **username_0**: thanks @HerrK. , corrected  

---

## 1 answers

### Answer by username_2 on 2024-10-04. Score: 2 | Accepted answer

Your definition of WARP needs correction.
The Walrasian demand function $x(p,w)$ satisfies the weak axiom of revealed preference if the following property holds:
$(\forall (p,w), (p',w'))(( p\cdot x(p',w')\leq w \ \wedge \ x(p',w')\neq x(p,w))\Rightarrow p'\cdot x(p,w)>w')$

The negation of WARP can be written as the following property:
$(\exists (p,w), (p',w'))(( p\cdot x(p',w')\leq w \ \wedge \ x(p',w')\neq x(p,w))\wedge p'\cdot x(p,w)\leq w')$

> **Comments:**
> **username_0**: Thanks, corrected. Could you help me with same definition but in context of choice structure:  If for some $B \in \mathcal{B}$ with $x,y \in B$ we have $x \in C(B)$, then for any $B' \in \mathcal{B}$ with $x,y \in B'$ and $y \in \ C(B')$, we must also have $x \in C(B')$  
> **username_2**: Do you want its negation?  
> **username_0**: yes  , because i want to know what conditions come under antecedent and for WARP to be vacuously the antecedent must include conditions upto ...$y \in C(B')$  
> **username_2**: Here is the negation of WARP: There exist $B,B'\in\mathcal{B}$ and there exist $x,y\in B\cap B'$ such that $x\in C(B)$, $y\in C(B')$ but $x\notin C(B')$.  
> **username_2**: Definition of WARP is $(\forall B, B'\in\mathcal{B})(\forall x,y\in B\cap B')((x\in C(B)\wedge y\in C(B'))\Rightarrow x\in C(B'))$  
> **username_0**: Thanks alot @Amit Sir  

---

