<p>This is a special case of Theorem 1 of [Richter, Marcel K. &quot;<a href="https://doi.org/10.2307/1909773" rel="nofollow noreferrer">Revealed preference theory.</a>&quot; <em>Econometrica</em> (1966): 635-645.]</p>

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2024-05-13T13:22:13.013Z
2024-05-13T13:22:13.013Z
Michael Greinecker
<p>Note this is question is not about consumer demand with price and income data.</p>
<p>This is a question about general choice theory. For reference, see: <a href="https://www.jstor.org/stable/2550390" rel="nofollow noreferrer">https://www.jstor.org/stable/2550390</a></p>
<p>See Debreu's Definition 4 and condition C1.</p>
<p><span class="math-container">$B$</span> is a menu set. <span class="math-container">$C(B)$</span> is a choice function with unique choice for each budget.</p>
<p>Definition: (direct revealed preference) we say <span class="math-container">$xPy$</span> if <span class="math-container">$y\in B$</span>, <span class="math-container">$x\in C(B)$</span>, and <span class="math-container">$x\neq y$</span>.</p>
<p>Definition: (indirect revealed preference) <span class="math-container">$xP^*y$</span> if there exists a sequence <span class="math-container">$x_1,...,x_n$</span> and <span class="math-container">$B_1,...,B_n$</span> such that <span class="math-container">$x\in C(B)$</span>, <span class="math-container">$x_1\in B$</span>, <span class="math-container">$x_1\in C(B_1)$</span>, <span class="math-container">$x_2\in B_1$</span>... <span class="math-container">$x_n\in C(B_n)$</span> and <span class="math-container">$y\in B_n$</span>.</p>
<p><span class="math-container">$SARP$</span>: <span class="math-container">$xP^*y$</span> implies not <span class="math-container">$yPx$</span>.</p>
<blockquote>
<p>To be proved: if <span class="math-container">$C$</span> satisfies <span class="math-container">$SARP$</span> then it is rationalizaed by a preference relation.</p>
</blockquote>
<p>I tried to search the literature but I find no proof for this general version.</p>

<microeconomics><consumer-theory><decision-theory>
Trying to find a proof for Strong Axiom of Revealed Preference with general choice set
High GPA
# Trying to find a proof for Strong Axiom of Revealed Preference with general choice set

(Asked by: username_0 on 2024-05-13)

Note this is question is not about consumer demand with price and income data.

This is a question about general choice theory. For reference, see: <https://www.jstor.org/stable/2550390>

See Debreu's Definition 4 and condition C1.

$B$ is a menu set. $C(B)$ is a choice function with unique choice for each budget.

Definition: (direct revealed preference) we say $xPy$ if $y\in B$, $x\in C(B)$, and $x\neq y$.

Definition: (indirect revealed preference) $xP^\*y$ if there exists a sequence $x\_1,...,x\_n$ and $B\_1,...,B\_n$ such that $x\in C(B)$, $x\_1\in B$, $x\_1\in C(B\_1)$, $x\_2\in B\_1$... $x\_n\in C(B\_n)$ and $y\in B\_n$.

$SARP$: $xP^\*y$ implies not $yPx$.

> To be proved: if $C$ satisfies $SARP$ then it is rationalizaed by a preference relation.

I tried to search the literature but I find no proof for this general version.

**Tags:** microeconomics, consumer-theory, decision-theory


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## 1 answers

### Answer by username_1 on 2024-05-13. Score: 1

This is a special case of Theorem 1 of [Richter, Marcel K. "[Revealed preference theory.](https://doi.org/10.2307/1909773)" *Econometrica* (1966): 635-645.]


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