<p>Let $C(y)$ denote the cost function and $y$ the quantity.</p>

<p>The average cost function is $AC=C/y$. This is increasing if its derivative is positive. Let $C'$ denote the derivative of $C$. The marginal cost function is $MC=C'$. We have:</p>

<p>$\frac{\partial AC}{\partial y}= \frac{C'y - C}{y^2}&gt;0$ by using the quotient rule. As $y^2&gt;0$, the condition becomes:</p>

<p>$C'y - C&gt;0$. </p>

<p>Rearranging yields:</p>

<p>$C' &gt; C/y$ </p>

<p>Hence: $MC&gt;AC$.</p>

CC BY-SA 3.0
2017-10-02T10:37:22.917Z
2017-10-02T11:51:18.423Z
2017-10-02T11:51:18.423Z
BB King
<p>We can calculate total cost as TC(Q)= Q*AC(Q) now just take derivative on both sides w.r.t Q so we get MC(Q)= AC(Q)+ Q * derivative of average cost w.r.t Q. Since Average cost is increasing function derivative must be non-negative so Marginal cost always greater than average cost.</p>

CC BY-SA 3.0
2017-10-02T11:10:16.970Z
2017-10-02T11:10:16.970Z
maths student
<p>I know the "common sense" proof, but how can we prove it algebraically?</p>

<microeconomics><production-function><cost><marginal>
Proof that for increasing AC MC(Q)>AC(Q) at any Q>0?
Alice
# Proof that for increasing AC MC(Q)>AC(Q) at any Q>0?

(Asked by: username_0 on 2017-10-02)

I know the "common sense" proof, but how can we prove it algebraically?

**Tags:** microeconomics, production-function, cost, marginal

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## 2 answers

### Answer by username_1 on 2017-10-02. Score: 2

Let $C(y)$ denote the cost function and $y$ the quantity.

The average cost function is $AC=C/y$. This is increasing if its derivative is positive. Let $C'$ denote the derivative of $C$. The marginal cost function is $MC=C'$. We have:

$\frac{\partial AC}{\partial y}= \frac{C'y - C}{y^2}>0$ by using the quotient rule. As $y^2>0$, the condition becomes:

$C'y - C>0$.

Rearranging yields:

$C' > C/y$

Hence: $MC>AC$.


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### Answer by username_2 on 2017-10-02. Score: 1

We can calculate total cost as TC(Q)= Q\*AC(Q) now just take derivative on both sides w.r.t Q so we get MC(Q)= AC(Q)+ Q \* derivative of average cost w.r.t Q. Since Average cost is increasing function derivative must be non-negative so Marginal cost always greater than average cost.


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