<p>In <em>Notes on the Theory of Choice</em> (assuming this is what you refer to by "Kreps (1988)"), Kreps does not appear to mention WARP as such. But he does refer to "Houthakker's axiom" (Houthakker 1950, <em>Economica</em>), which is actually SARP, not WARP. </p>

<p>Your conditions 1 and 2 are equivalent if there are only two goods. If there are more than two goods, WARP in condition 1 must be replaced by SARP to preserve the equivalence. Jehle and Reny (2011) have an intuitive explanation for this: </p>

<blockquote>
  <p>[W]ith two goods, the pairwise ranking of bundles implied through revealed preference turns out to have no intransitive cycles. [...] And when this is so, there will be a utility representation generating the choice function.</p>
</blockquote>

<p>SARP basically posits conditions that explicitly rule out intransitive cycles in revealed preferences with a general <span class="math-container">$n$</span>-good environment. </p>

<p>WARP/SARP differ from GARP in that the former presumes the availability of infinitely many choice data to build a choice <em>function</em> with, whereas GARP only requires a finite set of choice data. </p>

CC BY-SA 4.0
2019-05-14T22:23:29.003Z
2019-05-14T22:23:29.003Z
Herr K.
<p>Let <span class="math-container">$A$</span> be a menu and <span class="math-container">$R$</span> be a complete and transitive binary relation. Define choice correspondence generated by <span class="math-container">$R$</span>: </p>

<p><span class="math-container">$$c_R(A)=\{x\in A|| xRy \ \forall y\in A\}.$$</span></p>

<p>Theorem (from Kreps 1988): for any correspondence <span class="math-container">$c$</span>, the following conditions are equivalent:</p>

<ol>
<li><span class="math-container">$c$</span> satisfies WARP</li>
<li>There exists a transitive and complete <span class="math-container">$R$</span> such that <span class="math-container">$c=c_R$</span>.</li>
</ol>

<p>Given <span class="math-container">$A$</span> is finite and <span class="math-container">$c=c_R$</span>, it is obvious that <span class="math-container">$c$</span> can be represent by a utility function. Then why do we need SARP/GARP to rationalize a choice correspondence with a utility function? What was wrong with my logics?</p>

CC BY-SA 4.0
2019-05-14T21:35:02.783Z
Is Kreps (1988) the book titled *Notes on the Theory of Choice*? I don't think the book mentions WARP as such. Can you give a precise statement of the theorem in Kreps (1988)?
Herr K.
<microeconomics><mathematical-economics><utility><supply-and-demand><preferences>
WARP implies completeness, transitivity and thus rationalizability. What is wrong with the statement?
High GPA
# WARP implies completeness, transitivity and thus rationalizability. What is wrong with the statement?

(Asked by: username_0 on 2019-05-05)

Let $A$ be a menu and $R$ be a complete and transitive binary relation. Define choice correspondence generated by $R$:

$$c\_R(A)=\{x\in A|| xRy \ \forall y\in A\}.$$

Theorem (from Kreps 1988): for any correspondence $c$, the following conditions are equivalent:

1. $c$ satisfies WARP
2. There exists a transitive and complete $R$ such that $c=c\_R$.

Given $A$ is finite and $c=c\_R$, it is obvious that $c$ can be represent by a utility function. Then why do we need SARP/GARP to rationalize a choice correspondence with a utility function? What was wrong with my logics?

**Tags:** microeconomics, mathematical-economics, utility, supply-and-demand, preferences

> **Comments:**
> **username_1**: Is Kreps (1988) the book titled *Notes on the Theory of Choice*? I don't think the book mentions WARP as such. Can you give a precise statement of the theorem in Kreps (1988)?  

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## 1 answers

### Answer by username_1 on 2019-05-14. Score: 2 | Accepted answer

In *Notes on the Theory of Choice* (assuming this is what you refer to by "Kreps (1988)"), Kreps does not appear to mention WARP as such. But he does refer to "Houthakker's axiom" (Houthakker 1950, *Economica*), which is actually SARP, not WARP.

Your conditions 1 and 2 are equivalent if there are only two goods. If there are more than two goods, WARP in condition 1 must be replaced by SARP to preserve the equivalence. Jehle and Reny (2011) have an intuitive explanation for this:

> [W]ith two goods, the pairwise ranking of bundles implied through revealed preference turns out to have no intransitive cycles. [...] And when this is so, there will be a utility representation generating the choice function.

SARP basically posits conditions that explicitly rule out intransitive cycles in revealed preferences with a general $n$-good environment.

WARP/SARP differ from GARP in that the former presumes the availability of infinitely many choice data to build a choice *function* with, whereas GARP only requires a finite set of choice data.


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