<p>Suppose the total product is given by $f(x)$. Thus, the average product is $$AP(x)=\frac{f(x)}{x}.$$</p>

<p>Differentiating:
$$AP'(x)=\frac{x f'(x)-f(x)}{x^2},$$</p>

<p>which has the same sign as $x f'(x)-f(x)$. This is definitely negative if $f''(x)&lt;0$ (i.e., if $f$ is concave).</p>

<p>But it can also be negative, even if $f$ is locally convex. Take a look at this figure:</p>

<p><a href="https://i.sstatic.net/uJf4i.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/uJf4i.png" alt="enter image description here"></a></p>

<p>At the point marked with a blue disc: the slope of the red line is $f'(x)$. The slope of the black line is $f(x)/x$. We can see geometrically that $f(x)/x&gt;f'(x)$, or (after rearranging) that $f'(x)x&lt;f(x)$.</p>

<p>Thus, at the point indicated by the blue disc we know that $AP'(x)&lt;0$. But at this point we can also see in the figure that $f''(x)&gt;0,$ implying that the marginal product is increasing.</p>

<p>Since we have a counter-example, <strong>we conclude that a decreasing marginal product is not a necessary condition for a decreasing average product</strong>.</p>

CC BY-SA 3.0
2017-02-06T08:16:44.223Z
2017-02-06T08:16:44.223Z
Ubiquitous
<p>Taking @Henry's comment because he is too lazy to type it in as answer:</p>

<p>As long as $f$ is twice differentiable (which is likely if $MP(x)$ exists) just the fact that $f(x)$ is concave implies $\dfrac{d^2f(x)}{dx^2} &lt; 0$. This is also the first differentiate of $MP(x)$, thus it is diminishing.</p>

<p>So all you need is concavity. The 'rule' you refer to has different implications.</p>

CC BY-SA 3.0
2017-01-05T17:42:16.387Z
I think I've got an answer- if the production function isn't concave then it can be horizontal at the end. So MP is 0 whilst AP is still diminishing.  If the production function is concave then you are perfectly right.
Dtomas
CC BY-SA 3.0
2017-01-05T17:46:50.767Z
@Dtomas If that is the answer, I do not know the question.
Giskard
CC BY-SA 3.0
2017-01-05T17:47:21.140Z
@Dtomas Also it seems we have both mixed up concavity and strict concavity.
Giskard
CC BY-SA 3.0
2017-01-05T17:12:46.373Z
2017-02-06T09:07:17.890Z
2017-02-06T09:07:17.890Z
Giskard
<p>I am a little confused by the phrasing of the question, but I'll take it to mean this: </p>

<blockquote>
  <p>Does diminishing average product imply that marginal product is also decreasing?</p>
</blockquote>

<p>It is easy to show, as the OP acknowledges, that (under differentiability assumptions) that the average product of an input is decreasing in that input if and only if the marginal product of the said input is below the average product.</p>

<p>It turns out that this is essentially all you can say about the implications of diminishing average product on marginal product, as Ubiquitous shows in a separate answer.</p>

<p>I want to offer an alternative, simpler (albeit somewhat extreme) counter-example.</p>

<p>Consider a production function $F$ whose marginal product for input $x$ is $0$:</p>

<p>$$ \text{MP}_x = \frac{\partial F}{\partial x} = 0 $$</p>

<p>It is easy to see that as long as $F &gt; 0$, $\text{AP}_x$ is decreasing in $x$. Hence it is possible to have a production function that exhibits diminishing average product but a non-decreasing marginal product.</p>

CC BY-SA 3.0
2017-01-06T13:43:55.540Z
2017-02-06T20:14:59.297Z
2017-02-06T20:14:59.297Z
Theoretical Economist
<p>We have the rule 'If the marginal is less than the average, then the average declines'.</p>

<p>So if our production function for an input x and output y is concave, does that mean that MP must be diminishing given AP is diminishing?</p>

CC BY-SA 3.0
2017-01-05T16:39:19.530Z
If the production function $y=f(x)$ is concave, i.e. $\frac{d^2y}{dx^2}=f''(x) \lt 0$, I would have thought that in itself would be equivalent to saying marginal product $MP=\frac{dy}{dx}=f'(x)$ was decreasing
Henry
CC BY-SA 3.0
2017-01-05T16:59:33.783Z
@Henry please post answers as answers. Read [this](http://meta.economics.stackexchange.com/a/1651/1601) on meta.
Giskard
CC BY-SA 3.0
2017-01-05T17:08:26.187Z
@denesp: since I ignored the first half of the question and challenged the last four words of the second half, I would regard what I said as not being an answer to the question asked.  You are always free to turn somebody else's comment into an answer yourself
Henry
<production-function>
Does the marginal product need to be diminishing to have a diminishing average product?
Dtomas
# Does the marginal product need to be diminishing to have a diminishing average product?

(Asked by: username_0 on 2017-01-05)

We have the rule 'If the marginal is less than the average, then the average declines'.

So if our production function for an input x and output y is concave, does that mean that MP must be diminishing given AP is diminishing?

**Tags:** production-function

> **Comments:**
> **username_1**: If the production function $y=f(x)$ is concave, i.e. $\frac{d^2y}{dx^2}=f''(x) \lt 0$, I would have thought that in itself would be equivalent to saying marginal product $MP=\frac{dy}{dx}=f'(x)$ was decreasing  
> **username_2**: @username_1 please post answers as answers. Read [this](http://meta.economics.stackexchange.com/a/1651/1601) on meta.  
> **username_1**: @denesp: since I ignored the first half of the question and challenged the last four words of the second half, I would regard what I said as not being an answer to the question asked.  You are always free to turn somebody else's comment into an answer yourself  

---

## 3 answers

### Answer by username_3 on 2017-02-06. Score: 3

Suppose the total product is given by $f(x)$. Thus, the average product is $$AP(x)=\frac{f(x)}{x}.$$

Differentiating:
$$AP'(x)=\frac{x f'(x)-f(x)}{x^2},$$

which has the same sign as $x f'(x)-f(x)$. This is definitely negative if $f''(x)<0$ (i.e., if $f$ is concave).

But it can also be negative, even if $f$ is locally convex. Take a look at this figure:

[![enter image description here](https://i.sstatic.net/uJf4i.png)](https://i.sstatic.net/uJf4i.png)

At the point marked with a blue disc: the slope of the red line is $f'(x)$. The slope of the black line is $f(x)/x$. We can see geometrically that $f(x)/x>f'(x)$, or (after rearranging) that $f'(x)x

Thus, at the point indicated by the blue disc we know that $AP'(x)<0$. But at this point we can also see in the figure that $f''(x)>0,$ implying that the marginal product is increasing.

Since we have a counter-example, **we conclude that a decreasing marginal product is not a necessary condition for a decreasing average product**.


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### Answer by username_2 on 2017-01-05. Score: 2

Taking @username_1's comment because he is too lazy to type it in as answer:

As long as $f$ is twice differentiable (which is likely if $MP(x)$ exists) just the fact that $f(x)$ is concave implies $\dfrac{d^2f(x)}{dx^2} < 0$. This is also the first differentiate of $MP(x)$, thus it is diminishing.

So all you need is concavity. The 'rule' you refer to has different implications.

> **Comments:**
> **username_0**: I think I've got an answer- if the production function isn't concave then it can be horizontal at the end. So MP is 0 whilst AP is still diminishing.  If the production function is concave then you are perfectly right.  
> **username_2**: @username_0 If that is the answer, I do not know the question.  
> **username_2**: @username_0 Also it seems we have both mixed up concavity and strict concavity.  

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### Answer by username_4 on 2017-01-06. Score: 1

I am a little confused by the phrasing of the question, but I'll take it to mean this:

> Does diminishing average product imply that marginal product is also decreasing?

It is easy to show, as the OP acknowledges, that (under differentiability assumptions) that the average product of an input is decreasing in that input if and only if the marginal product of the said input is below the average product.

It turns out that this is essentially all you can say about the implications of diminishing average product on marginal product, as username_3 shows in a separate answer.

I want to offer an alternative, simpler (albeit somewhat extreme) counter-example.

Consider a production function $F$ whose marginal product for input $x$ is $0$:

$$ \text{MP}\_x = \frac{\partial F}{\partial x} = 0 $$

It is easy to see that as long as $F > 0$, $\text{AP}\_x$ is decreasing in $x$. Hence it is possible to have a production function that exhibits diminishing average product but a non-decreasing marginal product.


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