<p>I could not add images in the comments, therefore I used this method.</p>

<p>I think that the convexity definition does not hold for the function, since there are pairs of points that when they are conected, produce a segment which are out of the set. Because of it, the set is not convex.</p>

<p>Here there is an example of such situation with this function, which does not show a convex set:
<a href="https://i.sstatic.net/r6pel.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/r6pel.png" alt="enter image description here"></a></p>

CC BY-SA 3.0
2016-12-13T23:13:25.853Z
2016-12-13T23:13:25.853Z
Ignasi Mata Pavia
<p>Is it possible to have an inferior good under the assumption of a utility function which is strictly monotonically increasing and strictly quasiconvex?</p>

<p>I do not think so, but I could not find a formal proof.</p>

<p>*My underlying idea is that whit an inferior good, there exist a point where the utility consuming less of one good is better and then its derivative negative at that point, moreove, I think that this behaviour would break the concavity shape.</p>

CC BY-SA 3.0
2016-12-12T21:53:49.363Z
I was reading those examples and they do not fulfill the convexity assumption.
Ignasi Mata Pavia
CC BY-SA 3.0
2016-12-12T21:59:02.500Z
Perhaps you have not spent enough time on the first example
$$
U(x,y) = \alpha_1 \ln(x-\gamma_x)- \alpha_2 \ln(\gamma_y - y)
$$
 which does fulfill all your conditions for $0 < \alpha_1 < \alpha_2$.
Giskard
CC BY-SA 3.0
2016-12-12T22:42:35.707Z
But the second derivative of U(x,y) over y is: alpha2/(gamma-y)^2 which is positive in its domain.
Ignasi Mata Pavia
CC BY-SA 3.0
2016-12-13T06:28:25.370Z
That is not the definition of convexity for multivariate functions. Please read the paper linked in the answer, it actually explains why convexity holds.
Giskard
<microeconomics><mathematical-economics>
Inferior goods, monotonic utility and strict concavity
Ignasi Mata Pavia
# Inferior goods, monotonic utility and strict concavity

(Asked by: username_0 on 2016-12-12)

Is it possible to have an inferior good under the assumption of a utility function which is strictly monotonically increasing and strictly quasiconvex?

I do not think so, but I could not find a formal proof.

\*My underlying idea is that whit an inferior good, there exist a point where the utility consuming less of one good is better and then its derivative negative at that point, moreove, I think that this behaviour would break the concavity shape.

**Tags:** microeconomics, mathematical-economics

> **Comments:**
> **username_0**: I was reading those examples and they do not fulfill the convexity assumption.  
> **username_1**: Perhaps you have not spent enough time on the first example $$ U(x,y) = \alpha_1 \ln(x-\gamma_x)- \alpha_2 \ln(\gamma_y - y) $$  which does fulfill all your conditions for $0 < \alpha_1 < \alpha_2$.  
> **username_0**: But the second derivative of U(x,y) over y is: alpha2/(gamma-y)^2 which is positive in its domain.  
> **username_1**: That is not the definition of convexity for multivariate functions. Please read the paper linked in the answer, it actually explains why convexity holds.  

---

## 1 answers

### Answer by username_0 on 2016-12-13. Score: 1

I could not add images in the comments, therefore I used this method.

I think that the convexity definition does not hold for the function, since there are pairs of points that when they are conected, produce a segment which are out of the set. Because of it, the set is not convex.

Here there is an example of such situation with this function, which does not show a convex set:
[![enter image description here](https://i.sstatic.net/r6pel.png)](https://i.sstatic.net/r6pel.png)


---

