<p>The paper [Reny, Philip J. &quot;<a href="https://doi.org/10.3982/ECTA12345" rel="nofollow noreferrer">A characterization of rationalizable consumer behavior.</a>&quot; <em>Econometrica</em> 83.1 (2015): 175-192.] answers the question. Here is the abstract:</p>
<blockquote>
<p>For an arbitrary data set <span class="math-container">$D =\{(p,x)\}\subseteq (\mathbb{R}_+^m\setminus\{0\})\times\mathbb{R}_+^m$</span>, finite or
infinite, it is shown that the following three conditions are
equivalent: (a) <span class="math-container">$D$</span> satisfies GARP; (b) <span class="math-container">$D$</span> can be rationalized by a
utility function; (c) <span class="math-container">$D$</span> can be rationalized by a utility function
that is quasiconcave, nondecreasing, and that strictly increases when
all its coordinates strictly increase. Examples of infinite data sets
satisfying GARP are provided for which every utility rationalization
fails to be lower semicontinuous, upper semicontinuous, or concave.
Thus condition (c) cannot be substantively improved upon.</p>
</blockquote>
<p>Reny specifies data sets without incomes; the way he defines it, a utility function <span class="math-container">$u$</span> rationalizes <span class="math-container">$D$</span> if for every <span class="math-container">$(p,x)\in D$</span>, <span class="math-container">$x$</span> maximizes <span class="math-container">$u$</span> among bundles that cost at most <span class="math-container">$p\cdot x$</span>. The data set does not allow for unspent money to be observed.</p>

CC BY-SA 4.0
2024-12-03T21:07:31.400Z
I truly appreciate your answer. The only problem is that there is no conitnuity in both condition (b) and (c).
High GPA
CC BY-SA 4.0
2024-12-03T21:08:32.997Z
I wonder if these two updated b',c' are equivalent: b')D
 can be rationalized by a continuous utility function; c') D is rationalized by a utility that is quasiconcave, nondecreasing, and continuous.
High GPA
CC BY-SA 4.0
2024-12-03T23:08:44.070Z
I'm not aware of such a result.
Michael Greinecker
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2024-12-02T23:51:56.617Z
2024-12-02T23:51:56.617Z
Michael Greinecker
<p>Here is the classic Afriat Theorem for revealed preference:</p>
<p>The following three statements are equivalent:</p>
<ol>
<li>The finite data set satisfies GARP</li>
<li>The finite data set is rationalized by a non-satiated utility function</li>
<li>The finite data set is rationalized by a concave, monotonic, continous utility</li>
</ol>
<p>I am interested in the condition 2 and 3. If we consider infinite data instead, the equivalence of 2 and 3 seem not hold.</p>
<p>I wonder if there is a similar result (for example, the equivalence of 2 and 3' where 3' is a modification of 3), for infinite data set.</p>
<p><a href="https://sites.pitt.edu/%7Eluca/ECON3030/class%208.pdf" rel="nofollow noreferrer">Reference:</a></p>
<p><a href="https://i.sstatic.net/1tqZWZ3L.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1tqZWZ3L.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2024-12-02T13:35:05.223Z
I don't think that 2 and 3 are equivalent even for the case of finite data set.
Amit
CC BY-SA 4.0
2024-12-02T16:09:07.767Z
@Amit Is right. Consider the utility function given by $u(x)=-\|x-b\|$ where $b$ is some bliss point and $\|\cdot\|$ the Euclidean norm.  If $p\cdot b<w$, then you can get the data point $(p,b,w)$, which clearly cannot be rationalized by a monotonic utility function.
Michael Greinecker
CC BY-SA 4.0
2024-12-02T23:03:19.380Z
@m Is this utility locally non-satiated? Sorry I forgot to include the reference.
High GPA
CC BY-SA 4.0
2024-12-02T23:04:11.303Z
@Amit Sorry forgot to include the "local-nonsatiation" in the theorem
High GPA
<microeconomics><theory><decision-theory><data>
Does the revealed preference result "equivalence of utility and concave utility" generalize to infinite data?
High GPA
# Does the revealed preference result "equivalence of utility and concave utility" generalize to infinite data?

(Asked by: username_0 on 2024-11-27)

Here is the classic Afriat Theorem for revealed preference:

The following three statements are equivalent:

1. The finite data set satisfies GARP
2. The finite data set is rationalized by a non-satiated utility function
3. The finite data set is rationalized by a concave, monotonic, continous utility

I am interested in the condition 2 and 3. If we consider infinite data instead, the equivalence of 2 and 3 seem not hold.

I wonder if there is a similar result (for example, the equivalence of 2 and 3' where 3' is a modification of 3), for infinite data set.

[Reference:](https://sites.pitt.edu/%7Eluca/ECON3030/class%208.pdf)

[![enter image description here](https://i.sstatic.net/1tqZWZ3L.png)](https://i.sstatic.net/1tqZWZ3L.png)

**Tags:** microeconomics, theory, decision-theory, data
> **Comments:**
> **username_1**: I don't think that 2 and 3 are equivalent even for the case of finite data set.  
> **username_2**: @Amit Is right. Consider the utility function given by $u(x)=-\|x-b\|$ where $b$ is some bliss point and $\|\cdot\|$ the Euclidean norm.  If $p\cdot b<w$, then you can get the data point $(p,b,w)$, which clearly cannot be rationalized by a monotonic utility function.  
> **username_0**: @m Is this utility locally non-satiated? Sorry I forgot to include the reference.  
> **username_0**: @Amit Sorry forgot to include the "local-nonsatiation" in the theorem  

---

## 1 answers

### Answer by username_2 on 2024-12-02. Score: 3 | Accepted answer

The paper [Reny, Philip J. "[A characterization of rationalizable consumer behavior.](https://doi.org/10.3982/ECTA12345)" *Econometrica* 83.1 (2015): 175-192.] answers the question. Here is the abstract:

> For an arbitrary data set $D =\{(p,x)\}\subseteq (\mathbb{R}\_+^m\setminus\{0\})\times\mathbb{R}\_+^m$, finite or
> infinite, it is shown that the following three conditions are
> equivalent: (a) $D$ satisfies GARP; (b) $D$ can be rationalized by a
> utility function; (c) $D$ can be rationalized by a utility function
> that is quasiconcave, nondecreasing, and that strictly increases when
> all its coordinates strictly increase. Examples of infinite data sets
> satisfying GARP are provided for which every utility rationalization
> fails to be lower semicontinuous, upper semicontinuous, or concave.
> Thus condition (c) cannot be substantively improved upon.

Reny specifies data sets without incomes; the way he defines it, a utility function $u$ rationalizes $D$ if for every $(p,x)\in D$, $x$ maximizes $u$ among bundles that cost at most $p\cdot x$. The data set does not allow for unspent money to be observed.

> **Comments:**
> **username_0**: I truly appreciate your answer. The only problem is that there is no conitnuity in both condition (b) and (c).  
> **username_0**: I wonder if these two updated b',c' are equivalent: b')D  can be rationalized by a continuous utility function; c') D is rationalized by a utility that is quasiconcave, nondecreasing, and continuous.  
> **username_2**: I'm not aware of such a result.  

---

