<p>(2) does not imply (1). Consider a utility function with "circular indifference curves", e.g. <span class="math-container">$u(x,y)=-(x-1)^2-(y-1)^2$</span>. At the bliss point <span class="math-container">$(1,1)$</span>, the function satisfies (2) but violates (1).</p>

<p>(2) does imply (3), and the proof should be similar to the one showing that (1) implies (3). </p>

CC BY-SA 4.0
2019-06-14T17:40:41.187Z
Herr's first example also shows that (3) does not imply (1), for example, if $x=(1,1)$.
Regio
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2021-04-20T00:08:01.957Z
@Regio You are right. Local non-satiation seems to be only a sufficient but not necessary condition for indifference "curve".
High GPA
CC BY-SA 4.0
2019-06-09T22:25:19.460Z
2019-06-09T22:25:19.460Z
Herr K.
<p>Let <span class="math-container">$u$</span> be a continuous utility function on <span class="math-container">$\mathbb R^2_+\setminus\{0\}$</span>. Consider the following three conditions:</p>

<ol>
<li>Local <strong>non satiation</strong> says that for any <span class="math-container">$x \in X$</span> and <span class="math-container">$\epsilon &gt; 0$</span>, there exists <span class="math-container">$y \in X$</span> such that <span class="math-container">$d(x,y) &lt; \epsilon$</span> and <span class="math-container">$U(x) &lt; U(y)$</span>.</li>
<li>Local non satiation* says that for any <span class="math-container">$x \in X$</span> and <span class="math-container">$\epsilon &gt; 0$</span>, there exists <span class="math-container">$y \in X$</span> such that <span class="math-container">$d(x,y) &lt; \epsilon$</span> and <span class="math-container">$U(x) \neq U(y)$</span>.</li>
<li>The indifference sets of <span class="math-container">$U$</span> are curves.</li>
</ol>

<p>As a standard result, (1) implies (3) and (3) implies(1). </p>

<p>Obviously, (1) implies (2) so (3) also implies (2). </p>

<p>Can (2) also imply (1) and (3)？</p>

<microeconomics><mathematical-economics><supply-and-demand><consumer-theory><decision-theory>
A weaker definition of local non-satiation can also imply indifference "curve"
High GPA
# A weaker definition of local non-satiation can also imply indifference "curve"

(Asked by: username_0 on 2019-06-09)

Let $u$ be a continuous utility function on $\mathbb R^2\_+\setminus\{0\}$. Consider the following three conditions:

1. Local **non satiation** says that for any $x \in X$ and $\epsilon > 0$, there exists $y \in X$ such that $d(x,y) < \epsilon$ and $U(x) < U(y)$.
2. Local non satiation\* says that for any $x \in X$ and $\epsilon > 0$, there exists $y \in X$ such that $d(x,y) < \epsilon$ and $U(x) \neq U(y)$.
3. The indifference sets of $U$ are curves.

As a standard result, (1) implies (3) and (3) implies(1).

Obviously, (1) implies (2) so (3) also implies (2).

Can (2) also imply (1) and (3)？

**Tags:** microeconomics, mathematical-economics, supply-and-demand, consumer-theory, decision-theory

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## 1 answers

### Answer by username_1 on 2019-06-09. Score: 4 | Accepted answer

(2) does not imply (1). Consider a utility function with "circular indifference curves", e.g. $u(x,y)=-(x-1)^2-(y-1)^2$. At the bliss point $(1,1)$, the function satisfies (2) but violates (1).

(2) does imply (3), and the proof should be similar to the one showing that (1) implies (3).

> **Comments:**
> **username_2**: Herr's first example also shows that (3) does not imply (1), for example, if $x=(1,1)$.  
> **username_0**: @Regio You are right. Local non-satiation seems to be only a sufficient but not necessary condition for indifference "curve".  

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