<p>The term &quot;convex to the origin&quot; seems to have been used in the prosaic meaning of &quot;bending towards the origin&quot; without a formal mathematical definition. The note</p>
<blockquote>
<p>Kozlik, Adolf. &quot;<a href="https://www.jstor.org/stable/1804240" rel="noreferrer">Note on the Terminology Convex and Concave.</a>&quot; <em>The
American Economic Review</em> 31.1 (1941): 103-105.</p>
</blockquote>
<p>discusses the variation of early usage, which was not uniform. In Hicks' &quot;<a href="https://archive.org/details/in.ernet.dli.2015.223876" rel="noreferrer">Value and Capital</a>,&quot; for example, the term refers to an explanation of what it means that the marginal rate of substitution varies in a certain way.</p>
<p>Generally, the term &quot;convex to the origin&quot; comes from a time when the standard of mathematical rigor in economic theory was very different. With the adoption of convex analysis, convexity properties of preferences have replaced such discussions of visual properties of indifference curves. While it might be possible to give reasonable mathematical definitions of &quot;convex to the origin,&quot; this is not how the term was originally used.</p>

CC BY-SA 4.0
2024-03-25T13:49:06.467Z
2024-03-25T13:49:06.467Z
Michael Greinecker
<p><a href="https://economics.stackexchange.com/q/25255/1601">This question</a> asks about what &quot;convex to origin&quot; means. IMO even though there are half a dozen answers, they are all unsatisfactory. So in this question I ask for a precise definition of &quot;convex to origin&quot;, a  <strong>reference</strong> (where did you get the definition).</p>

CC BY-SA 4.0
2024-03-24T23:59:42.520Z
It would be helpful to know what makes the other answers unsatisfactory and how you want the 5 utility functions to be categorized.
Michael Greinecker
CC BY-SA 4.0
2024-03-25T03:17:07.320Z
I think none of the answers provide an actual definition rather than a vague notion, and none provides a reference to a source which has a definition.
Giskard
CC BY-SA 4.0
2024-03-25T03:18:36.687Z
I want the utility functions categorized based on whether their *indifference curves (or some of them) are "convex to origin"* or not.
Giskard
CC BY-SA 4.0
2024-03-25T07:26:56.180Z
"Convex" is an English word with nonmathematical uses. Used that way, "convex to the origin" means the same yhing as "curved to the origin."
Michael Greinecker
CC BY-SA 4.0
2024-03-25T07:48:32.790Z
Well, I have no clue what curved to origin means, so ¯\\_(ツ)_/¯ If it is trivial, can you maybe translate it to math for me? Or are you saying that a mathematical definition is impossible?
Giskard
CC BY-SA 4.0
2024-03-25T07:52:02.227Z
What I'm saying is that when Hicks, Robinson, and Shepherd used the expression in their writing, they did not use it as a mathematical term.
Michael Greinecker
CC BY-SA 4.0
2024-03-25T07:56:58.797Z
So you don't think an explicit definition exists, and people may use it differently?
Giskard
CC BY-SA 4.0
2024-03-25T08:03:25.563Z
Kinda. There certainly seemed to be discussion about whether there is a uniform use: https://www.jstor.org/stable/1804240
Michael Greinecker
CC BY-SA 4.0
2024-03-25T08:04:52.860Z
Please post your final comment as an answer, it's great. IMO it would also be nice if you posted it with a slightly longer explanation under the other question, but that is up to you.
Giskard
CC BY-SA 4.0
2024-03-25T08:08:37.607Z
Okay, I'll think about how to phrase it appropriately.
Michael Greinecker
<microeconomics><utility><definition>
Convex to origin - precise definition
Giskard
# Convex to origin - precise definition

(Asked by: username_0 on 2024-03-24)

[This question](https://economics.stackexchange.com/q/25255/1601) asks about what "convex to origin" means. IMO even though there are half a dozen answers, they are all unsatisfactory. So in this question I ask for a precise definition of "convex to origin", a **reference** (where did you get the definition).

**Tags:** microeconomics, utility, definition
> **Comments:**
> **username_1**: It would be helpful to know what makes the other answers unsatisfactory and how you want the 5 utility functions to be categorized.  
> **username_0**: I think none of the answers provide an actual definition rather than a vague notion, and none provides a reference to a source which has a definition.  
> **username_0**: I want the utility functions categorized based on whether their *indifference curves (or some of them) are "convex to origin"* or not.  
> **username_1**: "Convex" is an English word with nonmathematical uses. Used that way, "convex to the origin" means the same yhing as "curved to the origin."  
> **username_0**: Well, I have no clue what curved to origin means, so ¯\\_(ツ)_/¯ If it is trivial, can you maybe translate it to math for me? Or are you saying that a mathematical definition is impossible?  
> **username_1**: What I'm saying is that when Hicks, Robinson, and Shepherd used the expression in their writing, they did not use it as a mathematical term.  
> **username_0**: So you don't think an explicit definition exists, and people may use it differently?  
> **username_1**: Kinda. There certainly seemed to be discussion about whether there is a uniform use: https://www.jstor.org/stable/1804240  
> **username_0**: Please post your final comment as an answer, it's great. IMO it would also be nice if you posted it with a slightly longer explanation under the other question, but that is up to you.  
> **username_1**: Okay, I'll think about how to phrase it appropriately.  

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## 1 answers

### Answer by username_1 on 2024-03-25. Score: 5 | Accepted answer

The term "convex to the origin" seems to have been used in the prosaic meaning of "bending towards the origin" without a formal mathematical definition. The note

> Kozlik, Adolf. "[Note on the Terminology Convex and Concave.](https://www.jstor.org/stable/1804240)" *The
> American Economic Review* 31.1 (1941): 103-105.

discusses the variation of early usage, which was not uniform. In Hicks' "[Value and Capital](https://archive.org/details/in.ernet.dli.2015.223876)," for example, the term refers to an explanation of what it means that the marginal rate of substitution varies in a certain way.

Generally, the term "convex to the origin" comes from a time when the standard of mathematical rigor in economic theory was very different. With the adoption of convex analysis, convexity properties of preferences have replaced such discussions of visual properties of indifference curves. While it might be possible to give reasonable mathematical definitions of "convex to the origin," this is not how the term was originally used.


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