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 $R^2$?

The only journal that published it is behind a paywall: http://www.jstor.org/stable/2296210?seq=1#page_scan_tab_contents but check your school library access for it.

The proof is pretty complicated and is based on an induction argument. When I've tried to link SARP and WARP, I've only ever found references to his paper.


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.


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