<p>No. Any constant utility function is concave and, hence, quasi-concave and is representing preferences that are as far from being locally non-satiated as can be.</p>
<p>However, strict quasi-concavity (and, therefore, also strict concavity) together with continuity implies that preferences can have at most one satiation-point and that local non-satiation (whichis defined in terms of a for-all-statement) only fails there. To see this, note first that if there were two global satiation points, then any proper convex combination would be even better, which is impossible. To see that local non-satiation can't fail anywhere else,  note that any movement towards a better bundle must be an improvement by strong convexity of preferences (which followsfrom strict convexity and continuity.)</p>
<p>It is worth pointing out that the first welfare theorem holds when preferences are locally non-satiated with the possible exception of one satiation point.</p>

CC BY-SA 4.0
2022-06-12T10:26:59.197Z
A constant utility function satisfying $u(x) = c$ for any $x$, corresponds to global non-satiation, which implies (and is quite close to) local non-satiation.
Bertrand
CC BY-SA 4.0
2022-06-12T10:27:54.597Z
Uhm, not really.
Michael Greinecker
CC BY-SA 4.0
2022-06-12T12:43:46.493Z
Yes, sorry, it is global satiation....
Bertrand
CC BY-SA 4.0
2022-06-12T09:57:56.900Z
2022-06-12T09:57:56.900Z
Michael Greinecker
<p>Even strict concavity of utility function does not imply local non-satiation. For example: <span class="math-container">$u(x, y) = -x^2 - y^2$</span>. It is strictly concave because <span class="math-container">$x^2 + y^2$</span> is strictly convex. It does not satisfy local non-satiation because <span class="math-container">$(0,0)$</span> is the bliss point.</p>

CC BY-SA 4.0
2022-12-13T11:16:47.857Z
How do I check if a given utility function is locally non-satiated, say U = X1/X2 or U = X1-X2
reindeer
CC BY-SA 4.0
2022-06-12T11:30:33.460Z
2022-06-12T11:30:33.460Z
Amit
<p>I got a utility function with convex Indifferences curves and therefore convex preferences.
Convexity of preferences implies Quasi-concavity. I would like to know if there is a relation between convexity of the preferences and Local non satiation.
Does Concavity or quasi-concavity imply local non satiation?</p>
<p>Thank you</p>

<preferences>
Does Concavity or quasi-concavity imply local non satiation?
dewewdew
# Does Concavity or quasi-concavity imply local non satiation?

(Asked by: username_0 on 2022-06-12)

I got a utility function with convex Indifferences curves and therefore convex preferences.
Convexity of preferences implies Quasi-concavity. I would like to know if there is a relation between convexity of the preferences and Local non satiation.
Does Concavity or quasi-concavity imply local non satiation?

Thank you

**Tags:** preferences


---

## 2 answers

### Answer by username_3 on 2022-06-12. Score: 4 | Accepted answer

Even strict concavity of utility function does not imply local non-satiation. For example: $u(x, y) = -x^2 - y^2$. It is strictly concave because $x^2 + y^2$ is strictly convex. It does not satisfy local non-satiation because $(0,0)$ is the bliss point.

> **Comments:**
> **username_4**: How do I check if a given utility function is locally non-satiated, say U = X1/X2 or U = X1-X2  

---

### Answer by username_1 on 2022-06-12. Score: 1

No. Any constant utility function is concave and, hence, quasi-concave and is representing preferences that are as far from being locally non-satiated as can be.

However, strict quasi-concavity (and, therefore, also strict concavity) together with continuity implies that preferences can have at most one satiation-point and that local non-satiation (whichis defined in terms of a for-all-statement) only fails there. To see this, note first that if there were two global satiation points, then any proper convex combination would be even better, which is impossible. To see that local non-satiation can't fail anywhere else, note that any movement towards a better bundle must be an improvement by strong convexity of preferences (which followsfrom strict convexity and continuity.)

It is worth pointing out that the first welfare theorem holds when preferences are locally non-satiated with the possible exception of one satiation point.

> **Comments:**
> **username_2**: A constant utility function satisfying $u(x) = c$ for any $x$, corresponds to global non-satiation, which implies (and is quite close to) local non-satiation.  
> **username_1**: Uhm, not really.  
> **username_2**: Yes, sorry, it is global satiation....  

---

