<p>That WARP is equivalent to SARP for two goods is a result from <a href="https://doi.org/10.2307/2296210" rel="nofollow noreferrer">Rose (1958), &quot;Consistency of Preference: The Two-Commodity Case&quot;</a>. Although the proof is not so difficult is is rather too lengthy to put down here.</p>
<p>A few years ago, I wrote a paper (Transitivity of preferences: When does it matter?) that gives the exact conditions under which WARP and SARP are equivalent. You can find the published version <a href="https://econtheory.org/ojs/index.php/te/article/viewFile/20181043/21815/644" rel="nofollow noreferrer">here</a>. A necessary and sufficient condition turns out to be what we call a triangle condition. It boils down to the requirement that for any three prices vectors, one of the three price vectors should be below or above a linear combination of the other two price vectors. This is always satisfied in the two commodity case. Again the proof is not very complicated but still rather too lengthy to replicate here.</p>

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2021-09-12T07:42:27.517Z
2021-09-12T07:42:27.517Z
tdm
<p>I am considering an example where there are two goods and three budget sets <span class="math-container">$(\mathbf{p}^{(n)},w^{(n)}),n=1,2,3$</span>. If we assume <span class="math-container">$\mathbf{p}^{(n)} \cdot \mathbf{x}(\mathbf{p}^{(n+1)},w^{(n+1)}) \leq w^{(n)}$</span> for <span class="math-container">$n=1,2$</span>, and that WARP holds for any pair of bundles, how can I show that SARP holds?</p>
<p>Graphically, it is easy to convince myself that the statement holds, but I would like to explore a more rigorous proof.</p>

<microeconomics><preferences>
WARP implies SARP: A 2 Good Case
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# WARP implies SARP: A 2 Good Case

(Asked by: username_0 on 2021-09-12)

I am considering an example where there are two goods and three budget sets $(\mathbf{p}^{(n)},w^{(n)}),n=1,2,3$. If we assume $\mathbf{p}^{(n)} \cdot \mathbf{x}(\mathbf{p}^{(n+1)},w^{(n+1)}) \leq w^{(n)}$ for $n=1,2$, and that WARP holds for any pair of bundles, how can I show that SARP holds?

Graphically, it is easy to convince myself that the statement holds, but I would like to explore a more rigorous proof.

**Tags:** microeconomics, preferences

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## 1 answers

### Answer by username_1 on 2021-09-12. Score: 4 | Accepted answer

That WARP is equivalent to SARP for two goods is a result from [Rose (1958), "Consistency of Preference: The Two-Commodity Case"](https://doi.org/10.2307/2296210). Although the proof is not so difficult is is rather too lengthy to put down here.

A few years ago, I wrote a paper (Transitivity of preferences: When does it matter?) that gives the exact conditions under which WARP and SARP are equivalent. You can find the published version [here](https://econtheory.org/ojs/index.php/te/article/viewFile/20181043/21815/644). A necessary and sufficient condition turns out to be what we call a triangle condition. It boils down to the requirement that for any three prices vectors, one of the three price vectors should be below or above a linear combination of the other two price vectors. This is always satisfied in the two commodity case. Again the proof is not very complicated but still rather too lengthy to replicate here.


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