# How does this demand function satisfy the Weak axiom if it doesnt satisfy the necessary and sufficient conditions?

Let $x_k = \frac{w}{\sum{p_l}}$ for $k = 1...L$ be the demand function.

This demand function does satisfy the weak axiom and this can be shown simply.

The slutsky matrix is an $L \times L$ matrix of $0$'s for any (p,w). In Mas-Colell Microeconomic Theory page 35 and also in this publication - https://www.jstor.org/stable/1911539 , necessary and sufficient condition for the weak axiom to hold are that (i) S(p,w) is Negative semi-definite and (ii) $v^TS(p,w)v < 0$ whenever $v \neq \alpha p$ for any scalar $\alpha$, $v \neq 0$.

Clearly (i) is satisfied for this demand function and slutsky matrix. However, consider the price vector $[1,1,1,...1]$. The slutsky matrix is still the matrix of zeros. Consider $v=[2,1,1...1]$. $v^TS(p,w)v = 0$ which is not less than 0. Note that $v$ is not proportional to $p$. This appears to violate (ii). Where have I gone wrong?

This is exercise 2.F.17 from Mas-Colell Microeconomic theory.

1. If $x(p,w)$ satisfies Walras' law, homogeneity of degree zero, and the weak axiom, then the Slutsky matrix is negative semidefinite, that is, $v \cdot S(p,w) v \leq 0$ for any vector $v$.
2. If the Slutsky matrix is negative definite for all vectors $v$ that are not proportional to $p$, that is, if $v \cdot S(p,w) v < 0$ for such vectors $v$, and if $x(p,w)$ satisfies Walras' law and homogeneity of degree zero, then it satisfies the weak axiom.