<p>Think of &quot;quasiconcave&quot; as <em>single-peak</em>, and &quot;quasiconvex&quot; as <em>single-trough</em>. A function that has a single-peak has a unique maximum/supremum (both local and global), and a function that has a single-trough has a unique minimum/infimum (again both local and global). Clearly the two do not contradict one another. The production function is assumed to be quasiconcave and not necessarily quasiconvex because we're more interested how it peaks, i.e. reaches a maximum as in revenue maximization, than how it reaches a minimum.</p>

CC BY-SA 4.0
2022-08-17T01:21:35.887Z
Thank you for your reply. Is my two graphs also quasilinear, they seems also fit both difinition $f(tx+(1-t)y)\leq max\{f(x),f(y)\}$ and $f(tx+(1-t)y)\geq min\{f(x),f(y)\}$
LJNG
CC BY-SA 4.0
2022-08-17T02:07:42.237Z
@LJNG: Yes they are quasilinear in the sense defined in the [Wikipedia article](https://en.wikipedia.org/wiki/Quasiconvex_function#Definition_and_properties) you referred to. Note that quasilinearity is also commonly used in economics to refer a property of preferences/indifference curves; see [Quasilinear utility](https://en.wikipedia.org/wiki/Quasilinear_utility).
Herr K.
CC BY-SA 4.0
2022-08-17T05:56:51.953Z
@HerrK. Are quasilinear utility functions ever not quasilinear (QL) in this sense? What about vice-versa: can a function $U(x,y)$ that is QL in this sense always be transformed to a form of $v(x) + y$? Should I post this as a separate question?
Giskard
CC BY-SA 4.0
2022-08-17T11:17:48.817Z
@Giskard: These are different concepts, see https://math.stackexchange.com/questions/341991/relation-between-quasilinear-utility-and-quasilinear-function.
VARulle
CC BY-SA 4.0
2022-08-17T11:22:06.460Z
I disagree with thinking of quasiconcave as single-peaked. A monotonically increasing function as in the question admits no local maximum and is therefore not single-peaked, but any such function is quasiconcave. (Actually the concept of quasiconcavity is not very useful for increasing functions on a single dimension.)
VARulle
CC BY-SA 4.0
2022-08-17T16:50:17.417Z
@VARulle Thank you for your reply. What if we think of those two graphs are the projection(cross-section) of a $F(K, L)$ instead of $F(K)$, Are those $3D$ functions still a quasilinear function?
LJNG
CC BY-SA 4.0
2022-08-17T18:33:38.603Z
@VARulle: I did not intend to use single-peakedness as a rigorously defined property of functions, only as an intuitive notion for illustrative purposes. A monotonic but unbounded function does have a unique supremum in the extended reals ($\mathbb R\cup\{+\infty,-\infty\}$), and is thus "single-peaked" in that sense. This is in contrast to functions like $x^2$ or $\tan(x)$ which "explode" at multiple points on their domain.
Herr K.
CC BY-SA 4.0
2022-08-17T18:46:45.323Z
@Giskard: A quasilinear utility can be both quasiconcave and quasiconvex but not necessarily so. For example, a quasilinear utility $u(x,y)=x^2+y$ is not quasiconcave, though it is quasiconvex. I'm not sure if a utility function that is both quasiconcave and quasiconvex can always be transformed into a quasilinear form (via some positive monotonic function). Is this your question?
Herr K.
CC BY-SA 4.0
2022-08-17T18:56:23.217Z
That is the other half, yes.
Giskard
CC BY-SA 4.0
2022-08-17T21:57:52.253Z
@Giskard: That's a good question and I don't have a ready answer at the moment. Please do make a separate post on it.
Herr K.
CC BY-SA 4.0
2022-08-18T08:19:38.247Z
[Done](https://economics.stackexchange.com/questions/52411/quasicon-and-quasilinearity)!
Giskard
CC BY-SA 4.0
2022-08-18T08:20:30.643Z
Wow, I just figured out how to write comments shorter than 15 chars, very proud of myself.
Giskard
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2022-08-18T08:21:13.620Z
[1](https://economics.stackexchange.com/)
Giskard
CC BY-SA 4.0
2022-08-18T13:55:24.723Z
@Giskard: That's neat :)
Herr K.
CC BY-SA 4.0
2022-08-16T18:06:55.257Z
2022-08-16T18:06:55.257Z
Herr K.
<p><a href="https://i.sstatic.net/nMtUq.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/nMtUq.jpg" alt="enter image description here" /></a></p>
<p>Those two are generic production functions that we usually see, and I check the definition of quasiconvex and quasiconcave on wikipedia, and it seems those two graphs satisfy both definitions, and then so-called quasilinear. Then, I ponder why the production function we would generally assume to be quasiconcave only. Or I might misunderstand the definition or carelessly draw the wrong graph?</p>
<p>Appreciated any comment and help</p>

CC BY-SA 4.0
2022-08-16T15:56:00.417Z
Is it a function with a single variable, $F(K)$?
Giskard
CC BY-SA 4.0
2022-08-16T15:56:32.490Z
Is it also monotonically increasing?
Giskard
CC BY-SA 4.0
2022-08-16T16:03:53.687Z
@Giskard Thank you for your reply. Both Yes. But I have no idea why these questions are related to quasiconcavity. What happens if we suppose they are two variables function, let's considered it as a 3D picture. are they still quasi-convex?
LJNG
<microeconomics><mathematical-economics><economic-growth><production-function>
Please check those two production functions, they also seems to be quasiconvex?
LJNG
# Please check those two production functions, they also seems to be quasiconvex?

(Asked by: username_0 on 2022-08-16)

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

Those two are generic production functions that we usually see, and I check the definition of quasiconvex and quasiconcave on wikipedia, and it seems those two graphs satisfy both definitions, and then so-called quasilinear. Then, I ponder why the production function we would generally assume to be quasiconcave only. Or I might misunderstand the definition or carelessly draw the wrong graph?

Appreciated any comment and help

**Tags:** microeconomics, mathematical-economics, economic-growth, production-function
> **Comments:**
> **username_1**: Is it a function with a single variable, $F(K)$?  
> **username_1**: Is it also monotonically increasing?  
> **username_0**: @Giskard Thank you for your reply. Both Yes. But I have no idea why these questions are related to quasiconcavity. What happens if we suppose they are two variables function, let's considered it as a 3D picture. are they still quasi-convex?  

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## 1 answers

### Answer by username_2 on 2022-08-16. Score: 3 | Accepted answer

Think of "quasiconcave" as *single-peak*, and "quasiconvex" as *single-trough*. A function that has a single-peak has a unique maximum/supremum (both local and global), and a function that has a single-trough has a unique minimum/infimum (again both local and global). Clearly the two do not contradict one another. The production function is assumed to be quasiconcave and not necessarily quasiconvex because we're more interested how it peaks, i.e. reaches a maximum as in revenue maximization, than how it reaches a minimum.

> **Comments:**
> **username_0**: Thank you for your reply. Is my two graphs also quasilinear, they seems also fit both difinition $f(tx+(1-t)y)\leq max\{f(x),f(y)\}$ and $f(tx+(1-t)y)\geq min\{f(x),f(y)\}$  
> **username_2**: @LJNG: Yes they are quasilinear in the sense defined in the [Wikipedia article](https://en.wikipedia.org/wiki/Quasiconvex_function#Definition_and_properties) you referred to. Note that quasilinearity is also commonly used in economics to refer a property of preferences/indifference curves; see [Quasilinear utility](https://en.wikipedia.org/wiki/Quasilinear_utility).  
> **username_1**: @HerrK. Are quasilinear utility functions ever not quasilinear (QL) in this sense? What about vice-versa: can a function $U(x,y)$ that is QL in this sense always be transformed to a form of $v(x) + y$? Should I post this as a separate question?  
> **username_3**: @Giskard: These are different concepts, see https://math.stackexchange.com/questions/341991/relation-between-quasilinear-utility-and-quasilinear-function.  
> **username_3**: I disagree with thinking of quasiconcave as single-peaked. A monotonically increasing function as in the question admits no local maximum and is therefore not single-peaked, but any such function is quasiconcave. (Actually the concept of quasiconcavity is not very useful for increasing functions on a single dimension.)  
> **username_0**: @VARulle Thank you for your reply. What if we think of those two graphs are the projection(cross-section) of a $F(K, L)$ instead of $F(K)$, Are those $3D$ functions still a quasilinear function?  
> **username_2**: @VARulle: I did not intend to use single-peakedness as a rigorously defined property of functions, only as an intuitive notion for illustrative purposes. A monotonic but unbounded function does have a unique supremum in the extended reals ($\mathbb R\cup\{+\infty,-\infty\}$), and is thus "single-peaked" in that sense. This is in contrast to functions like $x^2$ or $\tan(x)$ which "explode" at multiple points on their domain.  
> **username_2**: @Giskard: A quasilinear utility can be both quasiconcave and quasiconvex but not necessarily so. For example, a quasilinear utility $u(x,y)=x^2+y$ is not quasiconcave, though it is quasiconvex. I'm not sure if a utility function that is both quasiconcave and quasiconvex can always be transformed into a quasilinear form (via some positive monotonic function). Is this your question?  
> **username_1**: That is the other half, yes.  
> **username_2**: @Giskard: That's a good question and I don't have a ready answer at the moment. Please do make a separate post on it.  
> **username_1**: [Done](https://economics.stackexchange.com/questions/52411/quasicon-and-quasilinearity)!  
> **username_1**: Wow, I just figured out how to write comments shorter than 15 chars, very proud of myself.  
> **username_1**: [1](https://economics.stackexchange.com/)  
> **username_2**: @Giskard: That's neat :)  

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