# Relationship between strong and weak axioms of revealed preference

I keep seeing these following facts just asserted while reading:

Let W = weak axiom of revealed preference Let S = strong axiom of revealed preference Let C = the commodity vector

1. $W \iff S$ when $C \in R^2$

2. $W \not\to S$ when $C \in R^i, i>2$

I can't find the 1958 paper by Rose that most other papers cite but I am interested in the proof for 1.

My thoughts about it:

I think that any agent whose demand struct satisfies W for a two-dimensional commodity space must have rational preferences. Since his preferences are rational, his demand structure must satisfy S. Is this roughly correct?

My questions: 1. Anyone have a reliable link to Rose's paper? 2. Anyone have a reliable link to any alternative sources?

1. If we are in $R^2$ and we have that xRy, is it true that the euclidean distance from the origin to x must be greater than the euclidean distance from the origin to y? If so, is it possible to use this property to show that $W\iff S$ in $R^2$?

Below just a sketch. Well, there is another way to prove this. First assume Walras' Law $p'x(p,w)=w$, second by WARP we know that the demand is Homogeneous of Degree Zero $x(\alpha p, \alpha w)=x(p,w)$. This is a result by John ( John, R. (2000). A first order characterization of generalized monotonicity. Mathematical Programming) , 88(1), 147–155. This means that the demand system L=2, can be normalized and that we can check only the Slutsky Matrix $S_{i,j}$ for $i,j \in {1,\cdots,L-1}$ after removing its last column and row, due to the smaller dimensionality product of the HD0 property and Walras' law. OK, so for L=2, the reduced Slutsky matrix is only a scalar, this is symmetric by default and by WARP it is also NSD, that means I can apply the results of integrability of the demand (basically any new proof of the original Hurwicz-Uzawa result, that dispenses with the wealth effects boundedness) to conclude that the demand system is generated by maximizing a continuously differentiable utility function $u$ subject to the linear budget constraint $p'x=w$ that is monotone. This in turn is enough to prove that the demand system satisfies GARP and if unique valued it satisfies SARP.