# Afriat theorem for negative goods

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.

• I think if you allow prices to be negative in GARP, then requiring $x\in\mathbb R_+$ is without loss of generality. – Herr K. May 2 '19 at 16:54
• @HerrK. You mean allowing $x\in\mathbb R$ and $p\in\mathbb R$? – High GPA May 2 '19 at 22:39
• I mean $x\in\mathbb R_+$ and $p\in \mathbb R$. – Herr K. May 2 '19 at 22:44