<p>"In the context of economics", it is neither - because convexity in economics means the same it does in math. If you redefine its domain, such that it's only computed for any $t &gt; 4.12$, this function's image is clearly convex: no line between two vectors inside it isn't entirely contained in the set.</p>

<p>P.S: Although convexity has the same definition in both fields, it has an important economic interpretation and is a very important hypothesis in many classical general equilibrium models.</p>

CC BY-SA 4.0
2018-09-16T05:10:53.237Z
2018-09-16T21:50:10.160Z
2018-09-16T21:50:10.160Z
Pedro Cavalcante
<p>Given the following cost function, where t is the quantity of some product.</p>

<p>$$C(t) = 1/3t^3 - 7t^2 +11t + 50$$</p>

<p>here is a graph between $t= 0$ and $t = 25$</p>

<p><a href="https://i.sstatic.net/Nqq6e.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Nqq6e.jpg" alt="enter image description here"></a></p>

<p>We are asked if this function is convex or concave?</p>

<p>Technically it is neither, but I am asking, <strong>in the context of economics,</strong> do we have to define intervals and say, between $0$ and $4.11$, it is concave, and for $t&gt;4.12$ we have a convex function?</p>

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2018-09-16T01:41:42.763Z
By the way, can you derive the TR function from this TC equation if you know the price is $10?
C. Master
CC BY-SA 4.0
2018-09-16T12:49:34.700Z
Do you mean $(1/3)t^3$?  If so the second derivative is $2t-14$ and the point of inflexion (change from concave to convex) is at $t$ such that $2t-14=0$.
Adam Bailey
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2018-09-16T23:19:07.450Z
@AdamBailey, Yes Adam, 1/3 t^3
C. Master
CC BY-SA 4.0
2018-09-16T23:20:53.903Z
But I am starting to think this cost function does not even make sense. I think it should be $111t$ instead of $11t$ in the third term. One of the reasons this doesn't make sense: How come the TC drops below the Fixed cost?
C. Master
<microeconomics><mathematical-economics><optimization>
Is this Cost function concave or convex?
C. Master
# Is this Cost function concave or convex?

(Asked by: username_0 on 2018-09-16)

Given the following cost function, where t is the quantity of some product.

$$C(t) = 1/3t^3 - 7t^2 +11t + 50$$

here is a graph between $t= 0$ and $t = 25$

[![enter image description here](https://i.sstatic.net/Nqq6e.jpg)](https://i.sstatic.net/Nqq6e.jpg)

We are asked if this function is convex or concave?

Technically it is neither, but I am asking, **in the context of economics,** do we have to define intervals and say, between $0$ and $4.11$, it is concave, and for $t>4.12$ we have a convex function?

**Tags:** microeconomics, mathematical-economics, optimization

> **Comments:**
> **username_0**: By the way, can you derive the TR function from this TC equation if you know the price is $10?  
> **username_1**: Do you mean $(1/3)t^3$?  If so the second derivative is $2t-14$ and the point of inflexion (change from concave to convex) is at $t$ such that $2t-14=0$.  
> **username_0**: @username_1, Yes username_1, 1/3 t^3  
> **username_0**: But I am starting to think this cost function does not even make sense. I think it should be $111t$ instead of $11t$ in the third term. One of the reasons this doesn't make sense: How come the TC drops below the Fixed cost?  

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## 1 answers

### Answer by username_2 on 2018-09-16. Score: 3

"In the context of economics", it is neither - because convexity in economics means the same it does in math. If you redefine its domain, such that it's only computed for any $t > 4.12$, this function's image is clearly convex: no line between two vectors inside it isn't entirely contained in the set.

P.S: Although convexity has the same definition in both fields, it has an important economic interpretation and is a very important hypothesis in many classical general equilibrium models.


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