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

Question

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.

Answer 1

Your equivalence result is incorrect. The correct results are:

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.

Therefore, there is no contradiction in the fact that your demand function satisfies the weak axiom while its Slutsky matrix is only negative semidefinite (and not negative definite).