<p>GARP and Afrait theorem assume that the alternative <span class="math-container">$x\in\mathbb R_+$</span> is always positive. In some economic contexts, such as financial choices, the attribute can be negative.</p>

<p>I wonder if we can naturally relax the <span class="math-container">$\mathbb R_+$</span> constraint. i.e. let <span class="math-container">$x\in\mathbb R$</span>,</p>

<p>GARP: if <span class="math-container">$p_i⋅x_{i+1}≤p_i⋅x_i$</span> for <span class="math-container">$i∈\{1,...,N−1\}$</span> and <span class="math-container">$p_N⋅x_1≤p_N⋅x_N$</span>, then those inequalities must be equalities.</p>

<p>At a first glance, the generalization to negative number seems plausible and not problematic.</p>

CC BY-SA 4.0
2019-05-02T16:54:36.403Z
I think if you allow prices to be negative in GARP, then requiring $x\in\mathbb R_+$ is without loss of generality.
Herr K.
CC BY-SA 4.0
2019-05-02T22:39:55.063Z
@HerrK. You mean allowing $x\in\mathbb R$ and $p\in\mathbb R$?
High GPA
CC BY-SA 4.0
2019-05-02T22:44:16.260Z
I mean $x\in\mathbb R_+$ and $p\in \mathbb R$.
Herr K.
<microeconomics><mathematical-economics><decision-theory><choice-theory><demand>
Afriat theorem for negative goods
High GPA
# Afriat theorem for negative goods

(Asked by: username_0 on 2019-05-02)

GARP and Afrait theorem assume that the alternative $x\in\mathbb R\_+$ is always positive. In some economic contexts, such as financial choices, the attribute can be negative.

I wonder if we can naturally relax the $\mathbb R\_+$ constraint. i.e. let $x\in\mathbb R$,

GARP: if $p\_i⋅x\_{i+1}≤p\_i⋅x\_i$ for $i∈\{1,...,N−1\}$ and $p\_N⋅x\_1≤p\_N⋅x\_N$, then those inequalities must be equalities.

At a first glance, the generalization to negative number seems plausible and not problematic.

**Tags:** microeconomics, mathematical-economics, decision-theory, choice-theory, demand
> **Comments:**
> **username_1**: I think if you allow prices to be negative in GARP, then requiring $x\in\mathbb R_+$ is without loss of generality.  
> **username_0**: @HerrK. You mean allowing $x\in\mathbb R$ and $p\in\mathbb R$?  
> **username_1**: I mean $x\in\mathbb R_+$ and $p\in \mathbb R$.  

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## 0 answers

