<p>The expression tries to convey a visual notion of convexity. It is "convex to the origin" in the sense that if we "stand" at the origin, the point <span class="math-container">$(0,0)$</span>, and "look towards" the graph, we will perceive it as convex.  </p>

<p>In contrast, if we stand "above" such a graph looking towards it, taking "our" position as being the origin, it will be concave. This "relativity" I suspect comes from physics and the arbitrariness of the origin of the coordinate system there.</p>

CC BY-SA 4.0
2018-10-30T09:10:59.797Z
What does it mean to be convex respective to one point?
John
CC BY-SA 4.0
2018-10-29T17:34:36.800Z
2018-10-29T17:34:36.800Z
Alecos Papadopoulos
<p>I think what people mean when they say "convex to the origin" (or to any point <span class="math-container">$p$</span>) is that the function is convex when looked at in a new basis, namely the basis resulting from a rotation such that the new x axis (call it x') is, up to a constant, tangent to the IC and the distance <span class="math-container">$|p-IC|$</span> is minimized by that point of tangency (<span class="math-container">$w$</span>).</p>

<p><a href="https://i.sstatic.net/drnYf.png" rel="noreferrer"><img src="https://i.sstatic.net/drnYf.png" alt="enter image description here"></a></p>

CC BY-SA 4.0
2018-10-30T17:22:45.010Z
This seems like the only answer that addresses the actual question. Can you please back up your definition with a reference?
Giskard
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2018-10-30T20:13:08.213Z
@denesp no because I made it up, that’s how I understand it but there’s no reason why this should be true
John
CC BY-SA 4.0
2018-10-30T09:38:27.280Z
2018-10-30T09:38:27.280Z
John
<p>I too have pondered this question. </p>

<p>I found a neat definition from the book 'Existence and Optimality of Competitive Equilibria' by Charalambos D. Aliprantis, Donald J. Brown, Owen (see page 9). I quote it here: </p>

<p>"Mathematically, a 'convex to the origin' curve is described by saying that if <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are any two points on the curve, then a ray passing through the origin <span class="math-container">$O$</span> and any point <span class="math-container">$X$</span> on the line segment <span class="math-container">$AB$</span> will meet the curve at most at one point <span class="math-container">$D$</span> between <span class="math-container">$O$</span> and <span class="math-container">$X$</span>". EDIT - this should read 'at exactly one point <span class="math-container">$D$</span> between <span class="math-container">$0$</span> and <span class="math-container">$X$</span> as opposed to at most one; the author appears to have made an error. </p>

<p><a href="https://i.sstatic.net/46RmD.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/46RmD.png" alt="Taken from the same source"></a></p>

<p>Under this definition, a curve shaped like the left-half of a U-shaped parabola, but never reaching a point at which its derivative is 0, is the type of nice convex to the origin indifference curve we dream about as economists. If we were to allow the slope of the indifference curve to become positive after some point then it will be possible to find draw a line from the origin that crosses two points of the curve, in which case the curve would not be convex to the origin. </p>

<p>When a utility function is a function of two variables x and y, an indifference curve is convex to the origin if the derivative of the indifference curves are always negative and the second derivatives are positive. That is, the indifference curves slope downwards always but the slope of the slope increases (from a negative value to a less negative value) as we move to the right. That is, the marginal rate of substitution (the negative of the slope of the indifference curve or equivalently, the magnitude of the slope of the indifference curve) decreases as we move to the right along an indifference curve. </p>

<p>Best, </p>

<p>Dutchman </p>

<p><strong><em>Edit</em></strong> in response to Giskard's comment: </p>

<p>Giskard is right, the definition would need to be amended to exactly one. </p>

CC BY-SA 4.0
2019-06-02T09:43:20.930Z
I don't get it. "a ray passing through the origin $O$ and any point $X$ on the line segment $AB$ will meet the curve at most at one point $D$ between $O$ and $X$".  Seems to me that this works even if the utility function is strictly concave. E.g. $U(x,y) = x^2 + y^2$ has this property. Perhaps if the definition was "exactly at one" and not "a most at one".
Giskard
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2019-06-02T10:29:31.460Z
@Giskard, I think that you are right. Thanks.
Dutchman
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2024-03-24T10:08:11.083Z
Come to think of it, a very squigly line could also have this property, so I still don't think this say anything about convexity?
Giskard
CC BY-SA 4.0
2019-05-31T09:54:35.423Z
2019-06-02T10:27:49.357Z
2019-06-02T10:27:49.357Z
Dutchman
<p>To discuss the convexity of the indifference curves for the given utility functions, we will use the approach of comparing the utility at the midpoint of two points on an indifference curve with the utility associated with those points. Let's graph the indifference curves for each function and analyze their convexity:</p>
<p>a. U(x, y) = min(x, y)</p>
<p>To begin, let's plot the indifference curves for this utility function:</p>
<pre><code>  y
  |
  |     /
  |    /
  |   /  
  |  /
  |_/_____________________
         x
</code></pre>
<p>In this case, the indifference curves represent the diagonal line y = x. Now, let's consider two arbitrary points (x1, y1) and (x2, y2) on the indifference curve y = x that promises U = k.</p>
<p>The midpoint of these two points is ((x1 + x2) / 2, (y1 + y2) / 2). Let's calculate the utility at this midpoint:</p>
<p>U((x1 + x2) / 2, (y1 + y2) / 2) = min(((x1 + x2) / 2), ((y1 + y2) / 2))</p>
<p>We need to compare this utility with k. Since the function U(x, y) = min(x, y) takes the smaller value between x and y, it is evident that min(((x1 + x2) / 2), ((y1 + y2) / 2)) will always be less than or equal to k. Therefore, the utility at the midpoint is at most equal to k.</p>
<p>Based on this analysis, we can conclude that the indifference curves for the utility function U(x, y) = min(x, y) are not convex.</p>

CC BY-SA 4.0
2023-06-15T11:12:48.693Z
2023-06-15T11:12:48.693Z
kashif zeb
<p>Note: this answer is more mathematically complicated than an undergraduate level, but I will try to give intuition as well. </p>

<p>I think it's easier to think about this problem in terms of preferences. A preference relation <span class="math-container">$\succsim$</span> over a set of alternatives <span class="math-container">$X$</span> is convex if 
<span class="math-container">$$x \succsim y \Rightarrow \lambda x + (1-\lambda) y \succsim y$$</span>
for every <span class="math-container">$\lambda \in (0,1]$</span>.</p>

<p>Basically, what this is saying is that if I like x more than y, I also like any bundle which is a mix between them better than y. So essentially, if I'm at y, then I'd like to get as close to x as possible with respect to y. </p>

<p>To visualize this, consider being at some value on an arbitrary indifference curve for convex preferences (e.g. Cobb-Douglass utility function). Pick another point which is on a strictly higher indifference curve. Then I can draw a straight line between the two bundles, and everything on that line is better than where I started. Graphically, this should be obvious, as everything between the two points should also be on a strictly higher indifference curve. Mathematically speaking, this is the same as saying that the upper contour set for x, <span class="math-container">$\{y : y \succsim x\}$</span> is a convex set for every x. </p>

CC BY-SA 4.0
2018-10-30T09:15:48.130Z
I like water better than fanta but fanta mixed with water is a disgrace and I'd rather drink just fanta. But I see your point
John
CC BY-SA 4.0
2018-10-30T17:19:41.030Z
"So essentially, if I'm at y, then I'd like to get as close to x as possible with respect to y." This is not true. Taking symmetric Cobb-Douglass preferences and $y = (0,1)$, $x = (1,0.01)$ I would much rather stay somewhere close to $\lambda = 1/2$ than get as close to $x$ as possible.
Giskard
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2018-10-30T17:20:43.353Z
Also the question is not about convexity, which is explained well in its first line, but about the concept of "convex to origin".
Giskard
CC BY-SA 4.0
2018-10-30T02:52:21.763Z
2018-10-30T02:52:21.763Z
NBm424
<p>There are two points to be noted here about the indifference curve:</p>
<ol>
<li>It is downward sloping =&gt; negative relation between 'Y' and 'X' =&gt; If 'Y' increases then 'X' decreases and vice-versa.</li>
<li>The magnitude of change(increase) in 'Y' wrt to a given change(decrease) in X =&gt; Nature of dy/dx.</li>
</ol>
<p>Conclusion:</p>
<ol>
<li>dy/dx is decreasing =&gt; d2y/dx2 &lt; 0 =&gt; law of Diminishing Marginal Rate of Substitution =&gt; A curve Convex to Origin
<a href="https://i.sstatic.net/1yDY9.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1yDY9.png" alt="enter image description here" /></a></li>
<li>dy/dx is constant =&gt; d2y/dx2 = 0 =&gt; the two commodities are perfect substitute=&gt; A straight line that neither convex or concave to origin</li>
<li>dy/dx is increasing =&gt; d2y/dx2 &gt; 0 =&gt; law of Increasing Marginal Rate of Substitution(generally does not happen) =&gt; A curve Concave to Origin</li>
</ol>

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2024-03-24T07:20:34.620Z
How do you reach a "conclusion" on a definition? What is your source for this interpretation?
Giskard
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2024-03-24T07:58:22.573Z
@Giskard I reached on the conclusions based on point#2 about that indifference curve. The source for the interpretation: https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://ncert.nic.in/textbook/pdf/leec202.pdf&ved=2ahUKEwiOoeuXtoyFAxWbxTgGHZaUAxsQFnoECA8QBg&usg=AOvVaw3MQNH_NlLVJZn6WB-inq1G
user5281371
CC BY-SA 4.0
2024-03-24T08:06:07.527Z
Thanks for the link, I will copy the relevant part (not a definition) here so others do not have to click it: "*It may be mentioned that the law of Diminishing Marginal Rate of Substitution causes an indifference curve to be convex to the origin.*" The word convex does not appear anywhere else in the linked text.
Giskard
CC BY-SA 4.0
2024-03-24T06:47:45.587Z
2024-03-24T06:52:22.437Z
2024-03-24T06:52:22.437Z
user5281371
<p>In case of any convex function, the value of the Arithmetic mean of any two points on the function is not less than the value of midpoint of the function in that interval.</p>

<p>Look at this function below:
<a href="https://i.sstatic.net/EhdVn.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/EhdVn.jpg" alt="enter image description here"></a></p>

<p>This function is convex to the point <span class="math-container">$(0,0)$</span> and concave to the point <span class="math-container">$(10,10)$</span>.</p>

CC BY-SA 4.0
2018-10-30T17:25:09.777Z
What you describe in your first sentence is the convex property, not the convex to origin property.
Giskard
CC BY-SA 4.0
2018-10-30T04:02:00.533Z
2018-11-01T00:27:07.913Z
2018-11-01T00:27:07.913Z
DrStrangeLove
<p>The convexity of an indifference curve results from the fact that the absolute value of its (negative) derivative, which is the marginal rate of substitution is decreasing. But why do we say that it's convex to the origin? </p>

<p>What is an implicit function that is convex and concave to the origin?</p>

<microeconomics>
"Convex to origin" indifference curves
John
# "Convex to origin" indifference curves

(Asked by: username_0 on 2018-10-29)

The convexity of an indifference curve results from the fact that the absolute value of its (negative) derivative, which is the marginal rate of substitution is decreasing. But why do we say that it's convex to the origin?

What is an implicit function that is convex and concave to the origin?

**Tags:** microeconomics

---

## 7 answers

### Answer by username_0 on 2018-10-30. Score: 5

I think what people mean when they say "convex to the origin" (or to any point $p$) is that the function is convex when looked at in a new basis, namely the basis resulting from a rotation such that the new x axis (call it x') is, up to a constant, tangent to the IC and the distance $|p-IC|$ is minimized by that point of tangency ($w$).

[![enter image description here](https://i.sstatic.net/drnYf.png)](https://i.sstatic.net/drnYf.png)

> **Comments:**
> **username_2**: This seems like the only answer that addresses the actual question. Can you please back up your definition with a reference?  
> **username_0**: @denesp no because I made it up, that’s how I understand it but there’s no reason why this should be true  

---

### Answer by username_1 on 2018-10-29. Score: 4

The expression tries to convey a visual notion of convexity. It is "convex to the origin" in the sense that if we "stand" at the origin, the point $(0,0)$, and "look towards" the graph, we will perceive it as convex.

In contrast, if we stand "above" such a graph looking towards it, taking "our" position as being the origin, it will be concave. This "relativity" I suspect comes from physics and the arbitrariness of the origin of the coordinate system there.

> **Comments:**
> **username_0**: What does it mean to be convex respective to one point?  

---

### Answer by username_3 on 2019-05-31. Score: 3

I too have pondered this question.

I found a neat definition from the book 'Existence and Optimality of Competitive Equilibria' by Charalambos D. Aliprantis, Donald J. Brown, Owen (see page 9). I quote it here:

"Mathematically, a 'convex to the origin' curve is described by saying that if $A$ and $B$ are any two points on the curve, then a ray passing through the origin $O$ and any point $X$ on the line segment $AB$ will meet the curve at most at one point $D$ between $O$ and $X$". EDIT - this should read 'at exactly one point $D$ between $0$ and $X$ as opposed to at most one; the author appears to have made an error.

[![Taken from the same source](https://i.sstatic.net/46RmD.png)](https://i.sstatic.net/46RmD.png)

Under this definition, a curve shaped like the left-half of a U-shaped parabola, but never reaching a point at which its derivative is 0, is the type of nice convex to the origin indifference curve we dream about as economists. If we were to allow the slope of the indifference curve to become positive after some point then it will be possible to find draw a line from the origin that crosses two points of the curve, in which case the curve would not be convex to the origin.

When a utility function is a function of two variables x and y, an indifference curve is convex to the origin if the derivative of the indifference curves are always negative and the second derivatives are positive. That is, the indifference curves slope downwards always but the slope of the slope increases (from a negative value to a less negative value) as we move to the right. That is, the marginal rate of substitution (the negative of the slope of the indifference curve or equivalently, the magnitude of the slope of the indifference curve) decreases as we move to the right along an indifference curve.

Best,

Dutchman

***Edit*** in response to Giskard's comment:

Giskard is right, the definition would need to be amended to exactly one.

> **Comments:**
> **username_2**: I don't get it. "a ray passing through the origin $O$ and any point $X$ on the line segment $AB$ will meet the curve at most at one point $D$ between $O$ and $X$".  Seems to me that this works even if the utility function is strictly concave. E.g. $U(x,y) = x^2 + y^2$ has this property. Perhaps if the definition was "exactly at one" and not "a most at one".  
> **username_3**: @Giskard, I think that you are right. Thanks.  
> **username_2**: Come to think of it, a very squigly line could also have this property, so I still don't think this say anything about convexity?  

---

### Answer by username_4 on 2023-06-15. Score: 0

To discuss the convexity of the indifference curves for the given utility functions, we will use the approach of comparing the utility at the midpoint of two points on an indifference curve with the utility associated with those points. Let's graph the indifference curves for each function and analyze their convexity:
a. U(x, y) = min(x, y)
To begin, let's plot the indifference curves for this utility function:
  y
  |
  |     /
  |    /
  |   /  
  |  /
  |_/_____________________
         x

In this case, the indifference curves represent the diagonal line y = x. Now, let's consider two arbitrary points (x1, y1) and (x2, y2) on the indifference curve y = x that promises U = k.
The midpoint of these two points is ((x1 + x2) / 2, (y1 + y2) / 2). Let's calculate the utility at this midpoint:
U((x1 + x2) / 2, (y1 + y2) / 2) = min(((x1 + x2) / 2), ((y1 + y2) / 2))
We need to compare this utility with k. Since the function U(x, y) = min(x, y) takes the smaller value between x and y, it is evident that min(((x1 + x2) / 2), ((y1 + y2) / 2)) will always be less than or equal to k. Therefore, the utility at the midpoint is at most equal to k.
Based on this analysis, we can conclude that the indifference curves for the utility function U(x, y) = min(x, y) are not convex.


---

### Answer by username_5 on 2018-10-30. Score: 0

Note: this answer is more mathematically complicated than an undergraduate level, but I will try to give intuition as well.

I think it's easier to think about this problem in terms of preferences. A preference relation $\succsim$ over a set of alternatives $X$ is convex if
$$x \succsim y \Rightarrow \lambda x + (1-\lambda) y \succsim y$$
for every $\lambda \in (0,1]$.

Basically, what this is saying is that if I like x more than y, I also like any bundle which is a mix between them better than y. So essentially, if I'm at y, then I'd like to get as close to x as possible with respect to y.

To visualize this, consider being at some value on an arbitrary indifference curve for convex preferences (e.g. Cobb-Douglass utility function). Pick another point which is on a strictly higher indifference curve. Then I can draw a straight line between the two bundles, and everything on that line is better than where I started. Graphically, this should be obvious, as everything between the two points should also be on a strictly higher indifference curve. Mathematically speaking, this is the same as saying that the upper contour set for x, $\{y : y \succsim x\}$ is a convex set for every x.

> **Comments:**
> **username_0**: I like water better than fanta but fanta mixed with water is a disgrace and I'd rather drink just fanta. But I see your point  
> **username_2**: "So essentially, if I'm at y, then I'd like to get as close to x as possible with respect to y." This is not true. Taking symmetric Cobb-Douglass preferences and $y = (0,1)$, $x = (1,0.01)$ I would much rather stay somewhere close to $\lambda = 1/2$ than get as close to $x$ as possible.  
> **username_2**: Also the question is not about convexity, which is explained well in its first line, but about the concept of "convex to origin".  

---

### Answer by username_6 on 2024-03-24. Score: 0

There are two points to be noted here about the indifference curve:

1. It is downward sloping => negative relation between 'Y' and 'X' => If 'Y' increases then 'X' decreases and vice-versa.
2. The magnitude of change(increase) in 'Y' wrt to a given change(decrease) in X => Nature of dy/dx.

Conclusion:

1. dy/dx is decreasing => d2y/dx2 < 0 => law of Diminishing Marginal Rate of Substitution => A curve Convex to Origin
   [![enter image description here](https://i.sstatic.net/1yDY9.png)](https://i.sstatic.net/1yDY9.png)
2. dy/dx is constant => d2y/dx2 = 0 => the two commodities are perfect substitute=> A straight line that neither convex or concave to origin
3. dy/dx is increasing => d2y/dx2 > 0 => law of Increasing Marginal Rate of Substitution(generally does not happen) => A curve Concave to Origin

> **Comments:**
> **username_2**: How do you reach a "conclusion" on a definition? What is your source for this interpretation?  
> **username_6**: @Giskard I reached on the conclusions based on point#2 about that indifference curve. The source for the interpretation: https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://ncert.nic.in/textbook/pdf/leec202.pdf&ved=2ahUKEwiOoeuXtoyFAxWbxTgGHZaUAxsQFnoECA8QBg&usg=AOvVaw3MQNH_NlLVJZn6WB-inq1G  
> **username_2**: Thanks for the link, I will copy the relevant part (not a definition) here so others do not have to click it: "*It may be mentioned that the law of Diminishing Marginal Rate of Substitution causes an indifference curve to be convex to the origin.*" The word convex does not appear anywhere else in the linked text.  

---

### Answer by username_7 on 2018-10-30. Score: 0

In case of any convex function, the value of the Arithmetic mean of any two points on the function is not less than the value of midpoint of the function in that interval.

Look at this function below:
[![enter image description here](https://i.sstatic.net/EhdVn.jpg)](https://i.sstatic.net/EhdVn.jpg)

This function is convex to the point $(0,0)$ and concave to the point $(10,10)$.

> **Comments:**
> **username_2**: What you describe in your first sentence is the convex property, not the convex to origin property.  

---

