<p>Hoping I don't confuse you any further.</p>

<p>Variable cost is the cost that depends on how much you produce (hence, variable). So in your case that would be:<br>
$\text{VC}:25Q-5Q^2+Q^3$</p>

<p>Now the average cost is the cost divided by how much is produced. The average variable cost then is:<br>
$\text{AVC} = \frac{\text{VC}}{Q} =25 - 5Q + Q^2$</p>

<p>Now you want the minimum of the average cost. You find the minimum by deriving:
$\frac{\partial \text{AVC}}{\partial Q} = 0;  - 5 + 2Q = 0$<br>
So you have that the minimum is at $Q=2.50$. Hurray. The AVC at the minimum is:<br>
$\text{AVC}(2.5) = 25-5*(2.5) + (2.5)^2$, that is <strong>18.75</strong>.</p>

<p>Now you also want the marginal cost, which is just:<br>
$\text{MC}:\frac{\partial \text{TC}}{\partial Q} = 25 - 10Q + 3 Q^2$<br>
$\text{MC}(2.5)=25-10*(2.5)+3*(2.5)^2$. And look at that we get back <strong>18.75</strong>.</p>

<p>So apparently they are the same. Who would have thought.</p>

CC BY-SA 3.0
2014-12-02T15:56:35.560Z
I think it would be much better not to answer questions like this, as it will only attract more of them.
Steven Landsburg
CC BY-SA 3.0
2014-12-02T16:10:11.330Z
To learn about Economics.SE's policy regarding answering homework questions, please refer to http://meta.economics.stackexchange.com/questions/24/how-should-we-deal-with-homework-questions
Martin Van der Linden
CC BY-SA 3.0
2014-12-02T17:08:34.663Z
Also visit the link posted by @MartinVanderLinden to influence that policy!
Ubiquitous
CC BY-SA 3.0
2014-12-03T03:15:38.663Z
sure, sure, let's keep these people out. We are already too many.
CarrKnight
CC BY-SA 3.0
2014-12-02T15:34:24.547Z
2014-12-02T15:34:24.547Z
CarrKnight
<p>I have lost all notes for this and can't seem to work it out, although i'm sure it is very simple I apologise in advance!</p>

<p>A manufacturer has the following short-run total cost function:</p>

<p>$$TC = 100 + 25Q – 5Q^2 + Q^3$$</p>

<p>Derive the average variable cost (AVC) function and show that, when AVC is a minimum, marginal cost (MC) is equal to AVC. </p>

CC BY-SA 3.0
2014-12-02T14:52:25.427Z
You don't need notes for this: Total Variable Cost is Total Cost minus Fixed Cost (and what is Fixed Cost?). Average Variable Cost is Variable Cost divided by the function's argument. Then, "minimum" and "marginal" both somehow relate to the mathematical concept of "derivative".
Alecos Papadopoulos
CC BY-SA 3.0
2014-12-02T14:59:24.350Z
I'm not given anything other than the short-run total cost function...
Charlie Robson
CC BY-SA 3.0
2014-12-02T15:32:07.433Z
Which is all you need. Have you never encountered a function minimization problem before? What strange kind of Microeconomics (or Business microeconomics) course are you attending?
Alecos Papadopoulos
<industrial-organisation>
Derive the average variable cost (AVC) function and show that, when AVC is a minimum, marginal cost (MC) is equal to AVC
Charlie Robson
# Derive the average variable cost (AVC) function and show that, when AVC is a minimum, marginal cost (MC) is equal to AVC

(Asked by: username_0 on 2014-12-02)

I have lost all notes for this and can't seem to work it out, although i'm sure it is very simple I apologise in advance!

A manufacturer has the following short-run total cost function:

$$TC = 100 + 25Q – 5Q^2 + Q^3$$

Derive the average variable cost (AVC) function and show that, when AVC is a minimum, marginal cost (MC) is equal to AVC.

**Tags:** industrial-organisation

> **Comments:**
> **username_1**: You don't need notes for this: Total Variable Cost is Total Cost minus Fixed Cost (and what is Fixed Cost?). Average Variable Cost is Variable Cost divided by the function's argument. Then, "minimum" and "marginal" both somehow relate to the mathematical concept of "derivative".  
> **username_0**: I'm not given anything other than the short-run total cost function...  
> **username_1**: Which is all you need. Have you never encountered a function minimization problem before? What strange kind of Microeconomics (or Business microeconomics) course are you attending?  

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## 1 answers

### Answer by username_2 on 2014-12-02. Score: 0 | Accepted answer

Hoping I don't confuse you any further.

Variable cost is the cost that depends on how much you produce (hence, variable). So in your case that would be:  
$\text{VC}:25Q-5Q^2+Q^3$

Now the average cost is the cost divided by how much is produced. The average variable cost then is:  
$\text{AVC} = \frac{\text{VC}}{Q} =25 - 5Q + Q^2$

Now you want the minimum of the average cost. You find the minimum by deriving:
$\frac{\partial \text{AVC}}{\partial Q} = 0; - 5 + 2Q = 0$  
So you have that the minimum is at $Q=2.50$. Hurray. The AVC at the minimum is:  
$\text{AVC}(2.5) = 25-5\*(2.5) + (2.5)^2$, that is **18.75**.

Now you also want the marginal cost, which is just:  
$\text{MC}:\frac{\partial \text{TC}}{\partial Q} = 25 - 10Q + 3 Q^2$  
$\text{MC}(2.5)=25-10\*(2.5)+3\*(2.5)^2$. And look at that we get back **18.75**.

So apparently they are the same. Who would have thought.

> **Comments:**
> **username_3**: I think it would be much better not to answer questions like this, as it will only attract more of them.  
> **username_4**: To learn about Economics.SE's policy regarding answering homework questions, please refer to http://meta.economics.stackexchange.com/questions/24/how-should-we-deal-with-homework-questions  
> **username_5**: Also visit the link posted by @username_4 to influence that policy!  
> **username_2**: sure, sure, let's keep these people out. We are already too many.  

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