<p>I'm doing an intermediate micro course, and we've been given a problem asking to draw curves that correspond to the utility function with the expression of a circle centred at (3,4). I understand that since the MRS of such an expression has &quot;x&quot; as the variable in the numerator, the MRS increases and the curves are concave towards the origin. A further question asks us to find whether the consumer will prefer more of the commodity to less of it.</p>
<p>On the one hand, it appears that the preferences of this consumer are not strictly monotonic, since increasing any one of the commodity is not preferred when the consumer's optimal bundle would be weighed towards any one of the commodities. On the other hand, finding partial derivatives of the equation with respect to &quot;x&quot; and &quot;y&quot; reveal that the commodities will only yield negative marginal utility in so far as x&lt;3 and y&lt;4. Would this not imply that the preferences become strictly monotonic after that, since the commodities behave like &quot;goods&quot; (that is, yield positive marginal utility?) Further, wouldn't that imply the indifference curves shouldn't be concave at all in the first place?</p>

CC BY-SA 4.0
2024-03-13T11:14:22.127Z
Hi! Can you please provide a definition of "*convex to origin*"? As seen [here](https://economics.stackexchange.com/questions/25255/convexity-of-indifference-curve) it is not clear what people mean by it. Also, have you considered that the utility functions $U(x,y)$ and $-U(x,y)$ have the exact same indifference curves?
Giskard
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2024-03-13T11:26:13.937Z
@Giskard I wasn't aware of that, but that clarifies things a bit. By convex I mean a standard well-behaved preference (we haven't done the mathematical treatment for convexity yet) but I think my confusion stemmed from the fact that I thought indifference curves can only be concave for two "bads". I suppose it is entirely possible for two "goods" (positive MU) to also have concave curves, such as in the case when a consumer doesn't prefer to consume them together. Is that right?
mayu
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2024-03-13T11:28:53.587Z
Hi! You don't write convex, you write "concave to origin". This is what I am asking about. Also concave/convex preferences are defined, concave/convex indifference curves (sets of points) seem unrelated.
Giskard
CC BY-SA 4.0
2024-03-13T11:30:29.737Z
@Giskard Ah, got it got it, apologies for the error!
mayu
CC BY-SA 4.0
2024-03-13T11:31:27.350Z
It is possible that the question has this concept, it is used in some places, but you need to include a definition if you want a rigorous answer.
Giskard
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2024-03-13T11:34:54.770Z
@Giskard yep, I was referring only to the shape of the indifference curves, so I think I should have clarified that. All we are taught right now regarding the convexity of preferences is that some utility functions do not translate into a convex indifference curves, and that some combinations of commodities, such as two bads, can be identified by indifference curves which are concave.
mayu
CC BY-SA 4.0
2024-03-13T11:39:24.927Z
"*two bads, can be identified by indifference curves which are concave*" Sorry, but it sounds like your professor is being imprecise or wrong :(
Giskard
CC BY-SA 4.0
2024-03-13T11:42:37.727Z
Again, the utility functions $x^2 + y^2$ and $-x^2 - y^2$ have the exact same indifference curves. In one, both goods are bads, in the other, both goods are goods. Yet the "concavity" is probably the same, since they are the same curves.
Giskard
CC BY-SA 4.0
2024-03-13T11:44:27.863Z
@Giskard yeah seems like my prof was a bit off, thanks!!
mayu
<utility><consumer-theory>
Monotonicity of Concave Indifference Curves
mayu
# Monotonicity of Concave Indifference Curves

(Asked by: username_0 on 2024-03-13)

I'm doing an intermediate micro course, and we've been given a problem asking to draw curves that correspond to the utility function with the expression of a circle centred at (3,4). I understand that since the MRS of such an expression has "x" as the variable in the numerator, the MRS increases and the curves are concave towards the origin. A further question asks us to find whether the consumer will prefer more of the commodity to less of it.

On the one hand, it appears that the preferences of this consumer are not strictly monotonic, since increasing any one of the commodity is not preferred when the consumer's optimal bundle would be weighed towards any one of the commodities. On the other hand, finding partial derivatives of the equation with respect to "x" and "y" reveal that the commodities will only yield negative marginal utility in so far as x<3 and y<4. Would this not imply that the preferences become strictly monotonic after that, since the commodities behave like "goods" (that is, yield positive marginal utility?) Further, wouldn't that imply the indifference curves shouldn't be concave at all in the first place?

**Tags:** utility, consumer-theory
> **Comments:**
> **username_1**: Hi! Can you please provide a definition of "*convex to origin*"? As seen [here](https://economics.stackexchange.com/questions/25255/convexity-of-indifference-curve) it is not clear what people mean by it. Also, have you considered that the utility functions $U(x,y)$ and $-U(x,y)$ have the exact same indifference curves?  
> **username_0**: @Giskard I wasn't aware of that, but that clarifies things a bit. By convex I mean a standard well-behaved preference (we haven't done the mathematical treatment for convexity yet) but I think my confusion stemmed from the fact that I thought indifference curves can only be concave for two "bads". I suppose it is entirely possible for two "goods" (positive MU) to also have concave curves, such as in the case when a consumer doesn't prefer to consume them together. Is that right?  
> **username_1**: Hi! You don't write convex, you write "concave to origin". This is what I am asking about. Also concave/convex preferences are defined, concave/convex indifference curves (sets of points) seem unrelated.  
> **username_0**: @Giskard Ah, got it got it, apologies for the error!  
> **username_1**: It is possible that the question has this concept, it is used in some places, but you need to include a definition if you want a rigorous answer.  
> **username_0**: @Giskard yep, I was referring only to the shape of the indifference curves, so I think I should have clarified that. All we are taught right now regarding the convexity of preferences is that some utility functions do not translate into a convex indifference curves, and that some combinations of commodities, such as two bads, can be identified by indifference curves which are concave.  
> **username_1**: "*two bads, can be identified by indifference curves which are concave*" Sorry, but it sounds like your professor is being imprecise or wrong :(  
> **username_1**: Again, the utility functions $x^2 + y^2$ and $-x^2 - y^2$ have the exact same indifference curves. In one, both goods are bads, in the other, both goods are goods. Yet the "concavity" is probably the same, since they are the same curves.  
> **username_0**: @Giskard yeah seems like my prof was a bit off, thanks!!  

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## 0 answers

