<p><strike>I disagree with @bbecon</strike>. I agree with @bbecon that concave utility functions present nice mathematical properties which help theorists develop analytical models.</p>
<p>If the OP's question was <em>why</em> utility functions are concave, the fundamental argument is that utility experiences <a href="https://en.wikipedia.org/wiki/Diminishing_returns" rel="noreferrer">diminishing returns</a>.</p>
<p>Examine the image below, taken form <a href="https://en.wikipedia.org/wiki/Budget_constraint" rel="noreferrer">here</a>.</p>
<p><a href="https://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Indifference_curves_showing_budget_line.svg/254px-Indifference_curves_showing_budget_line.svg.png" rel="noreferrer"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Indifference_curves_showing_budget_line.svg/254px-Indifference_curves_showing_budget_line.svg.png" alt="enter image description here" /></a></p>
<p>Let's say good <em>X</em> is coffee and good <em>Y</em> is cake. If you have 1 cup of coffee, getting a 2nd may increase your utility, but not by as much as the first one. Idem for 2 slices of cake. But if you could get 1 cup of coffee and 1 slice of cake, you would be most happy.</p>
<p>This argument fails under some special circumstances (e.g. <a href="https://en.wikipedia.org/wiki/Substitute_good" rel="noreferrer">this</a>), but I think it's reasonable in most cases.</p>

CC BY-SA 4.0
2021-08-02T09:55:28.963Z
This coupled with risk aversion makes more sense to me than bbecon's answer.
Richard Hardy
CC BY-SA 4.0
2021-08-02T11:16:04.693Z
@RichardHardy, risk aversion [probably doesn't exist](https://medium.com/incerto/the-logic-of-risk-taking-107bf41029d3)
crobar
CC BY-SA 4.0
2021-08-02T14:04:06.673Z
@crobar that article was rather lacking in logic and reasoning. I kept waiting for it to get better, figuring it must with the link text you used. But it didn't...
Eph
CC BY-SA 4.0
2021-08-02T14:20:10.153Z
@RichardHardy Classical risk aversion is the _result_ of diminishing returns. It appears whenever the utility function is [concave](https://en.wikipedia.org/wiki/Risk_aversion). It is not the _reason why_ utility functions are concave.
LBogaardt
CC BY-SA 4.0
2021-08-02T14:22:30.563Z
@Rick, if you want a more technical view try [this article](https://arxiv.org/abs/1906.04652), most 'risk aversion' can be attributed to incorrectly considering ergodicity. More background in a nature paper [here](https://www.nature.com/articles/s41567-019-0732-0)
crobar
CC BY-SA 4.0
2021-08-02T15:07:26.643Z
@crobar, I think by now Peters (2019) can be safely considered ignorable. I got this impression from the rebuttal by Doctor et al. (2019) and reactions elsewhere. Basically, it seems an enthusiastic physicist discovered economics and though he can fix it quickly and easily; not much of a surprise given [https://xkcd.com/793/](https://xkcd.com/793/). Just as unsurprisingly, his idea happens to be neither original nor very influential as it misses the target. But I am open to other perspectives (preferably with references).
Richard Hardy
CC BY-SA 4.0
2021-08-02T15:11:05.513Z
@crobar, regarding Taleb, I have my reservations about his mastery of economics and finance. The fact that he refers to Peters (2019) just happens to make them stronger.
Richard Hardy
CC BY-SA 4.0
2021-08-02T15:34:58.343Z
@RichardHardy, I guess this isn't the place for this debate, but Peters work is sound and backed by empirical evidence, see linked paper. Taleb is also correct, and has made a huge fortune in Finance by being correct.
crobar
CC BY-SA 4.0
2021-08-02T15:59:10.857Z
@crobar, Peters' (and Meder et al.) work may be mathematically sound but it breaks in the applications to economic problems; some of his applications/examples are not valid, other not new. Try [Doctor et al. (2019/2020)](https://www.nature.com/articles/s41567-020-01106-x) and especially their extended version / appendix (I have downloaded it but cannot find the source right now; should be possible to find from the paper itself) where all the details can be found. Perhaps your opinion of Peters (2019) will change after that.
Richard Hardy
CC BY-SA 4.0
2021-08-02T20:26:36.570Z
Indeed, not the place for this debate.
LBogaardt
CC BY-SA 4.0
2021-08-02T23:08:03.693Z
@crobar the idea is not sound, it is mathematically internally consistent but not applicable to economics. Geocentric model of solar system can also be mathematically made consistent and modeled if you willfully ignore wider research physics.
csilvia
CC BY-SA 4.0
2021-08-03T07:13:30.283Z
@crobar, I have opened [a new thread](https://economics.stackexchange.com/questions/47097/what-is-the-consensus-if-any-on-peters-the-ergodicity-problem-in-economics) to continue the debate on a meta level (not on the level of maths but rather reactions of respected economists and possible consensus in the profession). Consider opening another one if you would like to discuss Peters' critique on the "direct" (non-meta) level; I would be curious to see what the community thinks about it.
Richard Hardy
CC BY-SA 4.0
2021-08-03T07:15:57.867Z
@csilvia, I have opened [a new thread](https://economics.stackexchange.com/questions/47097) to continue the debate on Peters' critique.
Richard Hardy
CC BY-SA 4.0
2021-08-03T07:19:36.470Z
@LBogaardt, *Classical risk aversion is the result of diminishing returns.* Thanks for the comment. Took me a little while to think this through, but I guess you are right.
Richard Hardy
CC BY-SA 4.0
2021-08-03T10:22:17.707Z
@RichardHardy If you liked xkcd's treatment of physicists, you may also enjoy [this one from SMBC](https://www.smbc-comics.com/comic/2012-03-21).
Giskard
CC BY-SA 4.0
2021-08-01T17:43:58.503Z
2021-08-02T20:35:31.733Z
2021-08-02T20:35:31.733Z
LBogaardt
<p>More or less, yes.</p>
<p>Making the right assumption on the shape of the utility function allows you to prove existence or uniqueness of the equilibrium. The exact assumption you need depends on what exactly you are trying to prove and how general you want your result to be.</p>
<p>In the case of concavity, it also makes the equilibrium easier to find using the first-order conditions of the utility maximizer, because it makes sure that the local maximum that you find by setting the derivative of the Lagrangian to zero is also a global maximum.</p>

CC BY-SA 4.0
2021-08-01T20:18:35.670Z
You're welcome. Glad to be of help
bbecon
CC BY-SA 4.0
2021-08-02T12:03:16.007Z
-1 This may be correct as an explanation of why it can be convenient to assume that utility functions are concave.  But as an explanation of why concave utility functions are usually concave it is surely wrong - for that we need evidence from behaviour, eg of diminishing marginal rates of substitution.
Adam Bailey
CC BY-SA 4.0
2021-08-02T12:18:57.770Z
I believe my question was a bit misleading. I was interested in the mathematical reason why it is usually assumed that the utility function is concave. The issue of whether people actually have utility functions and what shape they are and why was not of my concern.
Ohad Osterreicher
CC BY-SA 4.0
2021-08-02T14:21:56.817Z
@OhadOsterreicher That's fine. In that case, this is indeed the correct answer. Could you update your original question above to make that a bit more clear (for future readers)?
LBogaardt
CC BY-SA 4.0
2021-08-02T19:16:11.870Z
Yes my interpretation of the question was "what does the concavity assumption buy you in practice", (he asked "why is it REQUIRED", which makes me think of a mathematical condition rather than an empirical property). Of course, that is a distinct question from whether concavity is a good assumption in the first place. One problem I have with LBogaardt comment is that utility functions express the same preferences up to a monotonic transformation. Ex: U=sqrt(xy) expresses the same (convex) preferences as U=(xy)^2, but the first utility is concave the latter convex.

https://bit.ly/3C7Nhmb
bbecon
CC BY-SA 4.0
2021-08-01T14:33:55.557Z
2021-08-01T14:33:55.557Z
bbecon
<p>Why is it usually required that utility function be concave? Is it because concavity is a necessary (or sufficient?) assumption for a unique equilibrium?</p>
<p>Can someone please spell this out for me?
Thank you.
Edit:
To clarify, I'm interested in the mathematical (modeling) reason for concavity. That concavity implies diminishing marginal utility and risk aversion is another matter.</p>

CC BY-SA 4.0
2021-08-01T19:28:14.010Z
In contexts with uncertainty, you can also connect concavity to risk aversion.
Bayesian
CC BY-SA 4.0
2021-08-01T20:41:41.010Z
I feel for me 'concavity' captures the spirit of a trade-off in economics. But yes, in terms of modelling, concave programming ensures a maximum.
EB3112
CC BY-SA 4.0
2021-08-02T18:54:01.347Z
I've taken the liberty of editing the title of your question to reflect what I understand you are asking.  Please feel free to reverse my edit if I've misunderstood.
Adam Bailey
<utility>
Why are utility functions typically assumed to be concave?
Ohad Osterreicher
# Why are utility functions typically assumed to be concave?

(Asked by: username_0 on 2021-08-01)

Why is it usually required that utility function be concave? Is it because concavity is a necessary (or sufficient?) assumption for a unique equilibrium?

Can someone please spell this out for me?
Thank you.
Edit:
To clarify, I'm interested in the mathematical (modeling) reason for concavity. That concavity implies diminishing marginal utility and risk aversion is another matter.

**Tags:** utility
> **Comments:**
> **username_1**: In contexts with uncertainty, you can also connect concavity to risk aversion.  
> **username_2**: I feel for me 'concavity' captures the spirit of a trade-off in economics. But yes, in terms of modelling, concave programming ensures a maximum.  
> **username_3**: I've taken the liberty of editing the title of your question to reflect what I understand you are asking.  Please feel free to reverse my edit if I've misunderstood.  

---

## 2 answers

### Answer by username_10 on 2021-08-01. Score: 7 | Accepted answer

More or less, yes.

Making the right assumption on the shape of the utility function allows you to prove existence or uniqueness of the equilibrium. The exact assumption you need depends on what exactly you are trying to prove and how general you want your result to be.

In the case of concavity, it also makes the equilibrium easier to find using the first-order conditions of the utility maximizer, because it makes sure that the local maximum that you find by setting the derivative of the Lagrangian to zero is also a global maximum.

> **Comments:**
> **username_10**: You're welcome. Glad to be of help  
> **username_3**: -1 This may be correct as an explanation of why it can be convenient to assume that utility functions are concave.  But as an explanation of why concave utility functions are usually concave it is surely wrong - for that we need evidence from behaviour, eg of diminishing marginal rates of substitution.  
> **username_0**: I believe my question was a bit misleading. I was interested in the mathematical reason why it is usually assumed that the utility function is concave. The issue of whether people actually have utility functions and what shape they are and why was not of my concern.  
> **username_4**: @OhadOsterreicher That's fine. In that case, this is indeed the correct answer. Could you update your original question above to make that a bit more clear (for future readers)?  
> **username_10**: Yes my interpretation of the question was "what does the concavity assumption buy you in practice", (he asked "why is it REQUIRED", which makes me think of a mathematical condition rather than an empirical property). Of course, that is a distinct question from whether concavity is a good assumption in the first place. One problem I have with LBogaardt comment is that utility functions express the same preferences up to a monotonic transformation. Ex: U=sqrt(xy) expresses the same (convex) preferences as U=(xy)^2, but the first utility is concave the latter convex.  https://bit.ly/3C7Nhmb  

---

### Answer by username_4 on 2021-08-01. Score: 14

I disagree with @bbecon. I agree with @bbecon that concave utility functions present nice mathematical properties which help theorists develop analytical models.

If the OP's question was *why* utility functions are concave, the fundamental argument is that utility experiences [diminishing returns](https://en.wikipedia.org/wiki/Diminishing_returns).

Examine the image below, taken form [here](https://en.wikipedia.org/wiki/Budget_constraint).

[![enter image description here](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Indifference_curves_showing_budget_line.svg/254px-Indifference_curves_showing_budget_line.svg.png)](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Indifference_curves_showing_budget_line.svg/254px-Indifference_curves_showing_budget_line.svg.png)

Let's say good *X* is coffee and good *Y* is cake. If you have 1 cup of coffee, getting a 2nd may increase your utility, but not by as much as the first one. Idem for 2 slices of cake. But if you could get 1 cup of coffee and 1 slice of cake, you would be most happy.

This argument fails under some special circumstances (e.g. [this](https://en.wikipedia.org/wiki/Substitute_good)), but I think it's reasonable in most cases.

> **Comments:**
> **username_5**: This coupled with risk aversion makes more sense to me than bbecon's answer.  
> **username_6**: @RichardHardy, risk aversion [probably doesn't exist](https://medium.com/incerto/the-logic-of-risk-taking-107bf41029d3)  
> **username_7**: @crobar that article was rather lacking in logic and reasoning. I kept waiting for it to get better, figuring it must with the link text you used. But it didn't...  
> **username_4**: @RichardHardy Classical risk aversion is the _result_ of diminishing returns. It appears whenever the utility function is [concave](https://en.wikipedia.org/wiki/Risk_aversion). It is not the _reason why_ utility functions are concave.  
> **username_6**: @Rick, if you want a more technical view try [this article](https://arxiv.org/abs/1906.04652), most 'risk aversion' can be attributed to incorrectly considering ergodicity. More background in a nature paper [here](https://www.nature.com/articles/s41567-019-0732-0)  
> **username_5**: @crobar, I think by now Peters (2019) can be safely considered ignorable. I got this impression from the rebuttal by Doctor et al. (2019) and reactions elsewhere. Basically, it seems an enthusiastic physicist discovered economics and though he can fix it quickly and easily; not much of a surprise given [https://xkcd.com/793/](https://xkcd.com/793/). Just as unsurprisingly, his idea happens to be neither original nor very influential as it misses the target. But I am open to other perspectives (preferably with references).  
> **username_5**: @crobar, regarding Taleb, I have my reservations about his mastery of economics and finance. The fact that he refers to Peters (2019) just happens to make them stronger.  
> **username_6**: @RichardHardy, I guess this isn't the place for this debate, but Peters work is sound and backed by empirical evidence, see linked paper. Taleb is also correct, and has made a huge fortune in Finance by being correct.  
> **username_5**: @crobar, Peters' (and Meder et al.) work may be mathematically sound but it breaks in the applications to economic problems; some of his applications/examples are not valid, other not new. Try [Doctor et al. (2019/2020)](https://www.nature.com/articles/s41567-020-01106-x) and especially their extended version / appendix (I have downloaded it but cannot find the source right now; should be possible to find from the paper itself) where all the details can be found. Perhaps your opinion of Peters (2019) will change after that.  
> **username_4**: Indeed, not the place for this debate.  
> **username_8**: @crobar the idea is not sound, it is mathematically internally consistent but not applicable to economics. Geocentric model of solar system can also be mathematically made consistent and modeled if you willfully ignore wider research physics.  
> **username_5**: @crobar, I have opened [a new thread](https://economics.stackexchange.com/questions/47097/what-is-the-consensus-if-any-on-peters-the-ergodicity-problem-in-economics) to continue the debate on a meta level (not on the level of maths but rather reactions of respected economists and possible consensus in the profession). Consider opening another one if you would like to discuss Peters' critique on the "direct" (non-meta) level; I would be curious to see what the community thinks about it.  
> **username_5**: @csilvia, I have opened [a new thread](https://economics.stackexchange.com/questions/47097) to continue the debate on Peters' critique.  
> **username_5**: @LBogaardt, *Classical risk aversion is the result of diminishing returns.* Thanks for the comment. Took me a little while to think this through, but I guess you are right.  
> **username_9**: @RichardHardy If you liked xkcd's treatment of physicists, you may also enjoy [this one from SMBC](https://www.smbc-comics.com/comic/2012-03-21).  

---

