<ul>
<li>In order to have an optimal solution, it is enought that preferences are continuous. (upper and lower contour sets are closed). It is however possible that the optimal solution is not unique.</li>
<li>If in addition indifference curves are convex, then the set of optimal solutions will be a convex set: if <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> are optimal solutions, then any convexi combination of the two will also be optimal.</li>
<li>If in addition indifference curves are strictly convex, then the optimal solution will be unique.</li>
<li>Differentiability plays no role in any of these results. The only reason why you would want the indifference curves to be differentiable is when you use first order conditions to derive the optimal choice. 

<ul>
<li>Differentiability (<span class="math-container">$C^1$</span> in particular) is enough to have smooth indifference curves. However, this does not guarantee that the optimal choice is unique. If the optimal choice is interior, then under differentiability, the first order conditions are satisfied. On the other hand, there might be other bundles that also satisfy the first order conditions but are not optimal (local minima or local optima). </li>
</ul></li>
</ul>

<p>I have no idea what TMS is.</p>

CC BY-SA 4.0
2020-05-26T09:27:36.330Z
TMS is often used for Adam Smith' *Theory of Moral Sentiments*. The connection is not immediate in this context, however...
VARulle
CC BY-SA 4.0
2020-05-26T09:38:14.163Z
Textbook models of consumer choice indeed sometimes need optimal solutions that can be derived from the FOCs, so they assume differentiability. But then you can weaken the strict convexity requirement to convexity.
VARulle
CC BY-SA 4.0
2020-05-26T13:36:57.783Z
Thanks for answering, @tdm. In translating my question I forgot to translate TMS. It's the MRS or marginal rate of substitution. 

Your answer doesn't seem to answer the main question (if it does and I missed it, I am sorry). Does the differentiability of the utility function alone guarantee there are no kinks in the indifference curves? Or are strictly convex preferences still required in order to have smooth indifference curves?
Pedro Cunha
CC BY-SA 4.0
2020-05-26T06:03:41.557Z
2020-05-27T06:04:14.650Z
2020-05-27T06:04:14.650Z
tdm
<p>Comment on the following affirmative:</p>

<p>In the traditional consumer model, the hypothesis of differentiability of the utility function and of convexity of preferences, assure the indifference curves have no kinks, without the need of the imposition of strict convexity for the preferences.</p>

<p>If the utility function is differentiable, then the indifference curve will have no kinks, seeing as how functions with kinks are not differentiable. I can't see what the importance of convexity is in this case. Convexity would interfere in the TMS and in the uniqueness of the optimal solution, right?</p>

<p>I've posted this question before, but it received a downvote (for which I couldn't figure out the reason), so I'm reposting in the hopes that it will be answered this time or at least someone will point out what the problem is with it so that I can fix it.</p>

<p>Thanks!</p>

CC BY-SA 4.0
2020-05-25T19:51:59.210Z
Hi! I am downvoting this question. Here is the reason: "please tell me if this is correct" type questions are a poor fit for the SE format.
Giskard
CC BY-SA 4.0
2020-05-25T19:52:11.007Z
Also, do not repost questions for any reason, that is spamming.
Giskard
CC BY-SA 4.0
2020-05-25T21:08:40.583Z
Ok. How would you suggest I ask it? Because for me at least it seems ok. I presented my thoughts, expressed my doubts ("I can't see what the importance of convexity is in this case. Convexity would interfere in the TMS and in the uniqueness of the optimal solution, right?") and opened it for discussion. 

As far as spamming goes, I've deleted the previous question. I should have simply edited it? I tried asking for clarification on what was wrong but got no replies.

Thanks
Pedro Cunha
CC BY-SA 4.0
2020-05-26T10:57:14.587Z
"*As far as spamming goes, I've deleted the previous question.*" So? This is still spamming. By pushing your question to the front of the queue again you are crowding out other questions. You should not make edits to old questions to push them to the front of the queue either.
Giskard
CC BY-SA 4.0
2020-05-26T13:22:15.913Z
So if my question is not answered for a long time or if it's downvoted without any reason being presented, I can't delete and write a new one nor can I edit the old one in order to fix whatever was wrong it? It seems to me then that the only choice is to give up on getting an answer for that particular question
Pedro Cunha
CC BY-SA 4.0
2020-05-26T17:54:47.880Z
You can always write a new question, but you should not repost your old question or edit it just for the sake of bumping it to the top of the queue. Seems like this was your motivation? "*so I'm reposting in the hopes that it will be answered this time*"
Giskard
CC BY-SA 4.0
2020-05-26T18:13:52.003Z
It wasn't for the sake of bumping it up. It had been downvoted and I didn't know why, which made impossible for me to fix whatever caused the downvote and made it much less likely to be answered. So I figured the best option would be to repost it, asking for clarification, hoping I could fix whatever was wrong and get an answer. I am sorry if it was a wrong thing to do; it didn't seem like it to me
Pedro Cunha
CC BY-SA 4.0
2020-05-26T19:18:47.640Z
I still do not understand why you would not ask for clarification under the original post. Because it would not get enough attention?
Giskard
CC BY-SA 4.0
2020-05-27T00:17:51.727Z
I did. It went unnoticed for several days.
Pedro Cunha
CC BY-SA 4.0
2020-05-27T06:07:52.293Z
To summarize, you did not repost it for the sake of bumping it up, you reposted it so that it would be on the front page and people would notice it more? Perhaps there are linguistic difficulties here, but I think these two things are the same? Again the problem is that by pushing your question to the front of the queue again you are crowding out other questions.
Giskard
CC BY-SA 4.0
2020-05-27T16:08:39.170Z
I reposted because I felt it was unfair to have my question disappear because of a downvote given without reason, which meant I couldn't even fix whatever was supposedly the problem (I tried asking for clarification on the question but that also got 0 replies). Anyway, I apologise for crowding out other questions. Won't happen again.
Pedro Cunha
<utility><preferences>
Differentiability of the utility function and indifference curves
Pedro Cunha
# Differentiability of the utility function and indifference curves

(Asked by: username_0 on 2020-05-25)

Comment on the following affirmative:

In the traditional consumer model, the hypothesis of differentiability of the utility function and of convexity of preferences, assure the indifference curves have no kinks, without the need of the imposition of strict convexity for the preferences.

If the utility function is differentiable, then the indifference curve will have no kinks, seeing as how functions with kinks are not differentiable. I can't see what the importance of convexity is in this case. Convexity would interfere in the TMS and in the uniqueness of the optimal solution, right?

I've posted this question before, but it received a downvote (for which I couldn't figure out the reason), so I'm reposting in the hopes that it will be answered this time or at least someone will point out what the problem is with it so that I can fix it.

Thanks!

**Tags:** utility, preferences
> **Comments:**
> **username_1**: Hi! I am downvoting this question. Here is the reason: "please tell me if this is correct" type questions are a poor fit for the SE format.  
> **username_1**: Also, do not repost questions for any reason, that is spamming.  
> **username_0**: Ok. How would you suggest I ask it? Because for me at least it seems ok. I presented my thoughts, expressed my doubts ("I can't see what the importance of convexity is in this case. Convexity would interfere in the TMS and in the uniqueness of the optimal solution, right?") and opened it for discussion.   As far as spamming goes, I've deleted the previous question. I should have simply edited it? I tried asking for clarification on what was wrong but got no replies.  Thanks  
> **username_1**: "*As far as spamming goes, I've deleted the previous question.*" So? This is still spamming. By pushing your question to the front of the queue again you are crowding out other questions. You should not make edits to old questions to push them to the front of the queue either.  
> **username_0**: So if my question is not answered for a long time or if it's downvoted without any reason being presented, I can't delete and write a new one nor can I edit the old one in order to fix whatever was wrong it? It seems to me then that the only choice is to give up on getting an answer for that particular question  
> **username_1**: You can always write a new question, but you should not repost your old question or edit it just for the sake of bumping it to the top of the queue. Seems like this was your motivation? "*so I'm reposting in the hopes that it will be answered this time*"  
> **username_0**: It wasn't for the sake of bumping it up. It had been downvoted and I didn't know why, which made impossible for me to fix whatever caused the downvote and made it much less likely to be answered. So I figured the best option would be to repost it, asking for clarification, hoping I could fix whatever was wrong and get an answer. I am sorry if it was a wrong thing to do; it didn't seem like it to me  
> **username_1**: I still do not understand why you would not ask for clarification under the original post. Because it would not get enough attention?  
> **username_0**: I did. It went unnoticed for several days.  
> **username_1**: To summarize, you did not repost it for the sake of bumping it up, you reposted it so that it would be on the front page and people would notice it more? Perhaps there are linguistic difficulties here, but I think these two things are the same? Again the problem is that by pushing your question to the front of the queue again you are crowding out other questions.  
> **username_0**: I reposted because I felt it was unfair to have my question disappear because of a downvote given without reason, which meant I couldn't even fix whatever was supposedly the problem (I tried asking for clarification on the question but that also got 0 replies). Anyway, I apologise for crowding out other questions. Won't happen again.  

---

## 1 answers

### Answer by username_2 on 2020-05-26. Score: 2 | Accepted answer

* In order to have an optimal solution, it is enought that preferences are continuous. (upper and lower contour sets are closed). It is however possible that the optimal solution is not unique.
* If in addition indifference curves are convex, then the set of optimal solutions will be a convex set: if $x\_1$ and $x\_2$ are optimal solutions, then any convexi combination of the two will also be optimal.
* If in addition indifference curves are strictly convex, then the optimal solution will be unique.
* Differentiability plays no role in any of these results. The only reason why you would want the indifference curves to be differentiable is when you use first order conditions to derive the optimal choice.
  + Differentiability ($C^1$ in particular) is enough to have smooth indifference curves. However, this does not guarantee that the optimal choice is unique. If the optimal choice is interior, then under differentiability, the first order conditions are satisfied. On the other hand, there might be other bundles that also satisfy the first order conditions but are not optimal (local minima or local optima).

I have no idea what TMS is.

> **Comments:**
> **username_3**: TMS is often used for Adam Smith' *Theory of Moral Sentiments*. The connection is not immediate in this context, however...  
> **username_3**: Textbook models of consumer choice indeed sometimes need optimal solutions that can be derived from the FOCs, so they assume differentiability. But then you can weaken the strict convexity requirement to convexity.  
> **username_0**: Thanks for answering, @tdm. In translating my question I forgot to translate TMS. It's the MRS or marginal rate of substitution.   Your answer doesn't seem to answer the main question (if it does and I missed it, I am sorry). Does the differentiability of the utility function alone guarantee there are no kinks in the indifference curves? Or are strictly convex preferences still required in order to have smooth indifference curves?  

---

