<p>Strict monotonicity of the (internal) utility function is not assumed but is implied by GARP thanks to <a href="http://www.pitt.edu/~luca/ECON2100/lecture_08.pdf" rel="nofollow noreferrer">Afriat's theorem</a></p>

<p>GARP is equivalent to:</p>

<blockquote>
  <p>There exists a continuous, strictly increasing, and concave utility function
  which rationalizes the data</p>
</blockquote>

<p>The equivalence does not hold for SARP. SARP implies GARP, but not vice-versa.</p>

<hr>

<p>Recall the actual definitions:</p>

<ul>
<li>Directly revealed weak preference; <span class="math-container">$x^a \ge^R x^b \; \text{if} \;p^ax^b \le w^a$</span></li>
<li>Indirectly revealed strict preference: 
<span class="math-container">$$ x^a &gt;^I x^b\; \text{if } \; \exists x^{i_1}, x^{i_2}, \ldots, x^{i_m} 
\; \text{such that} \; x^a \ge^R x^{i_1}, x^{i_1} \ge^R x^{i_2}, \ldots , x^{i_m} \ge^R x^b \\ \text{with at least one of these inequalities strict}$$</span></li>
<li><p>GARP: if <span class="math-container">$x^a \ge^R x^b$</span>, then not <span class="math-container">$x^b &gt;^I x^a$</span></p></li>
<li><p>an (internal) utility function <span class="math-container">$u:\mathbb{R}^n_+\to \mathbb{R}$</span> rationalizes the data <span class="math-container">$\{x^i, p^i, w^i\}_{i=1}^n$</span> if</p></li>
</ul>

<p><span class="math-container">$$ \forall x \in \mathbb{R}^n_+, \; p^ix \le w^i\; \text{implies}\; u(x^i) \ge u(x).$$</span> </p>

<p>So you can argue just "by inspection" that GARP is a "sham" set up just so we get a strict (internal) utility that rationalizes the data. (Why else ask for one of those inequalities to be strict? It's actually a key to the proof.) Apparently, the GARP axiom was formulated by Afriat himself, although <a href="https://journals.openedition.org/oeconomia/2742" rel="nofollow noreferrer">he called it</a> "cyclical consistency", so... draw your own conclusions how he may have come up with GARP.</p>

<p>The selling point of Afriat's result, as I understand it, is that you can check the "rational expectation" that the agent is utility maximizing (making optimal choices) by merely looking at a sample of its choices (i.e. testing this sample for GARP); furthermore, if the test is positive, you can also derive an utility function that is "as good as" the one that presumably generated the choices. There's an issue that the method is not robust to measurement error though.</p>

<blockquote>
  <blockquote>
    <p>The revealed preference principle of Samuelson … elaborated by Houthakker … easily gives a condition for the rejection of the hypothesis of existence. But the principle has been absent by which the hypothesis can be accepted or rejected on the basis of observed choice of the consumer, supposed to be finite in number; and, in the case of acceptance, a general method is needed for the actual construction of a utility function which will realize the hypothesis for the data. (Afriat, 1967, 68)</p>
    
    <p>Afriat’s approach … was truly constructive, offering an explicit algorithm to calculate a utility function consistent with the finite amount of data, whereas the other arguments were just existence proofs. This makes Afriat’s approach much more suitable as a basis of empirical analysis. Afriat’s approach was so novel that most researchers at the time did not recognize its value. (Varian, 2006, 101)</p>
  </blockquote>
  
  <p>[...] Houthakker had proven that if demand functions satisfied SARP there would always exist a rationalizing utility function, but provided no practical way to find it, while GARP techniques provided a way to construct such a utility function.</p>
</blockquote>

CC BY-SA 4.0
2019-04-27T05:04:33.963Z
But by defining the *directly revealed strict preference* as "$x^j\succ^Rx^k$ if $p^j\cdot x^k<w^j$", isn't monotonicity implicitly embedded in this definition? Since GARP would be violated if $p^j\cdot x^j<w^j$, we must have $w^j= p^j\cdot x^j$. This therefore suggests that $x^j\succ^Rx^k$ if $p^j\cdot x^k<p^j\cdot x^j$, or $x^j\ge x^k$ (element by element).
Herr K.
CC BY-SA 4.0
2019-04-27T05:33:20.077Z
@HerrK.: the GARP def uses the *directly revealed weak preference", not the strict one. On the other hand it does use the "indirectly revealed strict preference", so your could claim  Afriat's result is "obvious" because of the latter (proofs are not long), but that's a little subjective.
fantastic peace and prosperity
CC BY-SA 4.0
2019-04-27T00:06:41.030Z
2019-04-28T16:03:07.917Z
2019-04-28T16:03:07.917Z
fantastic peace and prosperity
<p>Monotonicity means the decision maker prefer more goods than less. It is not mentioned in textbook that SARP and GARP preasumed monotonicity, implicitly.</p>
<p>GARP: if <span class="math-container">$a$</span> is indirectly revealed preferred to <span class="math-container">$b$</span>, then <span class="math-container">$b$</span> is not directly revealed strictly preferred to <span class="math-container">$a$</span>.</p>
<p>SARP: if <span class="math-container">$a$</span> is indirectly revealed preferred to <span class="math-container">$b$</span>, and <span class="math-container">$a\neq b$</span>, then <span class="math-container">$b$</span> is not indirectly revealed preferred to <span class="math-container">$a$</span>.</p>
<p>Because <span class="math-container">$aRb$</span> (directly revealed) means <span class="math-container">$ap_a\geq bp_a$</span> and <span class="math-container">$aR_sb$</span> (directly revealed strictly) means <span class="math-container">$ap_a&gt;bp_a$</span>, does that implies monotonicty if we assume the price is positive?</p>

<microeconomics><consumer-theory><preferences><decision-theory><theory>
GARP and SARP assumed mononticity?
High GPA
# GARP and SARP assumed mononticity?

(Asked by: username_0 on 2019-04-25)

Monotonicity means the decision maker prefer more goods than less. It is not mentioned in textbook that SARP and GARP preasumed monotonicity, implicitly.

GARP: if $a$ is indirectly revealed preferred to $b$, then $b$ is not directly revealed strictly preferred to $a$.

SARP: if $a$ is indirectly revealed preferred to $b$, and $a\neq b$, then $b$ is not indirectly revealed preferred to $a$.

Because $aRb$ (directly revealed) means $ap\_a\geq bp\_a$ and $aR\_sb$ (directly revealed strictly) means $ap\_a>bp\_a$, does that implies monotonicty if we assume the price is positive?

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


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## 1 answers

### Answer by username_1 on 2019-04-27. Score: 4 | Accepted answer

Strict monotonicity of the (internal) utility function is not assumed but is implied by GARP thanks to [Afriat's theorem](http://www.pitt.edu/~luca/ECON2100/lecture_08.pdf)

GARP is equivalent to:

> There exists a continuous, strictly increasing, and concave utility function
> which rationalizes the data

The equivalence does not hold for SARP. SARP implies GARP, but not vice-versa.

---

Recall the actual definitions:

* Directly revealed weak preference; $x^a \ge^R x^b \; \text{if} \;p^ax^b \le w^a$
* Indirectly revealed strict preference:
  $$ x^a >^I x^b\; \text{if } \; \exists x^{i\_1}, x^{i\_2}, \ldots, x^{i\_m}
  \; \text{such that} \; x^a \ge^R x^{i\_1}, x^{i\_1} \ge^R x^{i\_2}, \ldots , x^{i\_m} \ge^R x^b \\ \text{with at least one of these inequalities strict}$$
* GARP: if $x^a \ge^R x^b$, then not $x^b >^I x^a$
* an (internal) utility function $u:\mathbb{R}^n\_+\to \mathbb{R}$ rationalizes the data $\{x^i, p^i, w^i\}\_{i=1}^n$ if

$$ \forall x \in \mathbb{R}^n\_+, \; p^ix \le w^i\; \text{implies}\; u(x^i) \ge u(x).$$

So you can argue just "by inspection" that GARP is a "sham" set up just so we get a strict (internal) utility that rationalizes the data. (Why else ask for one of those inequalities to be strict? It's actually a key to the proof.) Apparently, the GARP axiom was formulated by Afriat himself, although [he called it](https://journals.openedition.org/oeconomia/2742) "cyclical consistency", so... draw your own conclusions how he may have come up with GARP.

The selling point of Afriat's result, as I understand it, is that you can check the "rational expectation" that the agent is utility maximizing (making optimal choices) by merely looking at a sample of its choices (i.e. testing this sample for GARP); furthermore, if the test is positive, you can also derive an utility function that is "as good as" the one that presumably generated the choices. There's an issue that the method is not robust to measurement error though.

> > The revealed preference principle of Samuelson … elaborated by Houthakker … easily gives a condition for the rejection of the hypothesis of existence. But the principle has been absent by which the hypothesis can be accepted or rejected on the basis of observed choice of the consumer, supposed to be finite in number; and, in the case of acceptance, a general method is needed for the actual construction of a utility function which will realize the hypothesis for the data. (Afriat, 1967, 68)
> >
> > Afriat’s approach … was truly constructive, offering an explicit algorithm to calculate a utility function consistent with the finite amount of data, whereas the other arguments were just existence proofs. This makes Afriat’s approach much more suitable as a basis of empirical analysis. Afriat’s approach was so novel that most researchers at the time did not recognize its value. (Varian, 2006, 101)
>
> [...] Houthakker had proven that if demand functions satisfied SARP there would always exist a rationalizing utility function, but provided no practical way to find it, while GARP techniques provided a way to construct such a utility function.

> **Comments:**
> **username_2**: But by defining the *directly revealed strict preference* as "$x^j\succ^Rx^k$ if $p^j\cdot x^k<w^j$", isn't monotonicity implicitly embedded in this definition? Since GARP would be violated if $p^j\cdot x^j<w^j$, we must have $w^j= p^j\cdot x^j$. This therefore suggests that $x^j\succ^Rx^k$ if $p^j\cdot x^k<p^j\cdot x^j$, or $x^j\ge x^k$ (element by element).  
> **username_1**: @HerrK.: the GARP def uses the *directly revealed weak preference", not the strict one. On the other hand it does use the "indirectly revealed strict preference", so your could claim  Afriat's result is "obvious" because of the latter (proofs are not long), but that's a little subjective.  

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