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Maybe this is a well-known fact that I have somehow overlooked or forgotten etc., but I found myself curious recently if there are any conclusions about WARP holding over different dimensions for a given choice structure.

So, if I have a utility function, $N$ agents, $L$ commodities and price vectors in $R^L_+$, and I want to determine if WARP holds here, is it enough to assume that $L=2$ and prove WARP. Meaning, is it true that if WARP holds for $L=2$ it also holds for $L=N$, $N \in \mathbb{N}$

Edit : I feel like I should add that I know, generally, proving something for a specific case does not allow one to generalize the result. However, there are unique cases where such proofs can work. The first example that comes to mind is that whenever the excess demand function is homogeneous of degree zero we can reduce the problem to the simplex and it is sufficient to show that only n-1 markets clear.

Why I am asking?

I am trying to determine whether or not WARP holds for an adaptation of the (1999) Fehr & Schmidt Inequality Aversion utility function. I thought it might be easier to do by considering $L=2$

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    $\begingroup$ I think what you mean is: If fixing all but two variables in a finite consumption bundle, WARP holds, and this is true for any two variables, does WARP hold for all bundles? $\endgroup$ – Giskard Jun 14 '16 at 15:39
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I can shed some light on the question, but am not sure I can answer it as I am not sure it is really even well defined.

(1) The Weak Axiom of Revealed Preference is a decision theoretic concept regarding the choices of a single agent. So I do not understand how having $N$ agents is relevant to the problem.

(2) Generally speaking, if $U: X \to \mathbb{R}$ is a utility function and $\mathscr{C}$ is a choice correspondence over $X$ such that $\mathscr{C}(A) = \{x \in A \mid U(x) \geq U(y), \ \forall y \in A\}$ then $\mathscr{C}$ will satisfy WARP. This is a straightforward exercise in using the definitions, and the dimensionality of the space should play no role (its true over an abstract $X$).

(3) If the consumption space is $\mathbb{R}^l$ then preferences over defined $\mathbb{R}^l$. How you project preferences into $\mathbb{R}^2$ matters.

(4) If I interpret your question a la denesp, so that you ask: fix the $2 < k \leq l$ dimensions of consumption and only let dimensions 1 and 2 vary (of course, what we fix the other dimensions of still may make a difference), and assume that this restricted choice correspondence satisfied WARP will the choice correspondence in general satisfy WARP.

To answer (4): if the choices comes from a utility function then yes, trivially (see point (2)). If the choices are more general, then no. This fails severely. Take as a counter example a preference over $\mathbb{R}^3$ such that fixing the 3rd dimension, the consumer is indifferent over all elements (i.e., the first and second dimensions are null). WARP holds as $\mathscr{C}^{x}(A) = A$ for all $A \in \mathbb{R}^2 \times \{x\}$. This leaves the 3rd dimension wholly unrestricted. Letting the choice function over this last dimension be cyclic (i.e., fail WARP) and we see that the original choice would as well.

(5) What if we instead interpret your question as: we see every two dimensional projection of choice. (That is for every $i,j \leq l$, $i \neq j$, we see the projection of choices over the dimensions $i$ and $j$ fixing the other dimensions arbitrarily. Well, if the way we fix the other dimensions matters to the restricted choices (e.g., $\mathscr{C}^{x}$ over $\mathbb{R}^2 \times \{x\}$ is not the same as $\mathscr{C}^{y}$ over $\mathbb{R}^2 \times \{y\}$). Then we are back to the same type of problem as (5)---$\mathscr{C}$ could be cyclic when the dimensions are fixed differently.

What if how we fix the dimensions doesn't affect the restricted choice (so we have separable preferences, a la Koopmans). Then WARP will hold over $\mathscr{C}$ if it is rationalized by a preference relation. But, the result is not very interesting, since from the restricted choices know the choice of out of $\{x,y\}$ for all $x,y \in \mathbb{R}^l$. It is well known that only a binary choice is necessary to verify a rationalization. Without this additional constraint, I believe you could still cook up a counter example in the spirit of (4) where cyclically kicks in when certain elements (at least 3, pairwise distinct over 3 different dimensions) are present.

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