<p>Decreasing marginal returns to a factor means that keeping the other factors fixed, the marginal output generated by this factor is decreasing. When looking at returns to scale, we change all outputs. Increasing a factor with decreasing marginal returns can have an indirect effect in increasing the marginal productivity of other factors. If we increase all factors at the same time, the indirect effects may outweigh the direct effect. The production function $F:\mathbb{R}_+^2$ given by
$$F(x,y)=(x+1)^{2/3}(y+1)^{2/3}$$
has decreasing marginal factor productivities everywhere but not decreasing returns to scale (it doesn't have increasing returns to scale either).</p>

CC BY-SA 3.0
2017-09-24T22:51:46.297Z
2017-09-25T06:22:17.390Z
2017-09-25T06:22:17.390Z
Michael Greinecker
<p>For decreasing marginal returns we require second partial derivatives to be negative, since we examine what happens if we vary only one input</p>

<p>So any function 
$$y = \prod_{i=1}^{m}x_i^{a_i},\;\;\; 0&lt;a_i&lt;1\;\; \forall \,i$$ </p>

<p>with $\sum a_i &gt;1$ for increasing returns to scale, and with $\sum a_i &lt;1$ for decreasing returns to scale, since here we examine what happens of we increase by the same proportion all inputs.</p>

<hr>

<p><strong>Responding to comments</strong></p>

<p>We check returns to scale for this function by examining, for $k&gt;1$, the expression   </p>

<p>$$ \prod_{i=1}^{m}(kx_i)^{a_i} = k^{\sum a_i}\cdot y$$</p>

<p>If the sum of the alphas is higher than unity, output increases more than $k$ so we have increasing returns to scale, and correspondingly for deceasing returns to scale when the sum of the alphas is smaller than unity.  </p>

<p>Regarding decreasing marginal returns, the rate of change of marginal output generated by a factor keeping the others fixed is given by its own second partial derivative, which here is </p>

<p>$$\frac {\partial^2 y}{\partial x_i^2}= a_i(a_i-1)\cdot \frac {y}{x_i^2}$$</p>

<p>When $0&lt;a_i&lt;1$, these second partials are all negative.</p>

CC BY-SA 3.0
2017-09-25T05:47:21.203Z
What is your definition of decreasing/increasing returns to scale? For example, under the [definition](https://en.wikipedia.org/wiki/Returns_to_scale#Formal_definitions) used in the Wikipedia entry on returns to scale, a Cobb-Douglas production function cannot have positive or decreasing returns to scale (something the entry later gets wrong).
Michael Greinecker
CC BY-SA 3.0
2017-09-25T06:24:09.153Z
Moreover, this function doesn't have decreasing marginal returns everywhere, unless $m=1$.
Michael Greinecker
CC BY-SA 3.0
2017-09-25T06:46:59.737Z
@MichaelGreinecker I expanded my answer to answer your questions.
Alecos Papadopoulos
CC BY-SA 3.0
2017-09-25T06:50:05.507Z
You are ignoring the boundary, where some factors can be zero. In that case, the whole output is zero and remains so after doubling, and the marginal productivity of zero inputs is zero and therefore constant.
Michael Greinecker
CC BY-SA 3.0
2017-09-25T06:57:35.857Z
@MichaelGreinecker I certainly do, since it is a mathematical case which is rather uninteresting from the point of view of Economics. As regards the definition in wikipedia for the Cobb-Douglas function, it follows the convention to call "Cobb-Douglas" the function when the sum of exponents equals exactly unity, while when we want to use the exponential form with sum of exponents different than unity we usually call it "generalized Cobb-Douglas".
Alecos Papadopoulos
CC BY-SA 3.0
2017-09-25T07:00:18.047Z
I would like to see your definition of returns to scale that, unlike the definition used by the Wikipedia entry, exclude "mathematical case which is rather uninteresting from the point of view of Economics."
Michael Greinecker
CC BY-SA 3.0
2017-09-25T07:11:14.993Z
@MichaelGreinecker I am not sure that I understand what more should I write, except perhaps by adding the qualifier $x_i>0,\, \forall\, i$. All Economics textbooks I know don't even bother to mention that, exactly because a "zero-output case due to fixing an input to zero" is uninteresting for the purposes of Economics.
Alecos Papadopoulos
CC BY-SA 3.0
2017-09-25T07:40:24.187Z
I do hold the view that all economic textbooks can be wrong, in particular, those that make claims to mathematical rigor. A textbook that does not commit the mistake is "Microeconomic Foundations I" by David Kreps.
Michael Greinecker
CC BY-SA 3.0
2017-09-24T23:42:05.220Z
2017-09-25T06:46:57.570Z
2017-09-25T06:46:57.570Z
Alecos Papadopoulos
<p>I know that diminishing marginal returns even to all factors of production doesn't imply decreasing returns to scale. But could you please give me just an example of such production function?</p>

CC BY-SA 3.0
2017-09-24T23:30:08.290Z
In the title you write "increasing", in the question you write "decreasing". Correct and clarify.
Alecos Papadopoulos
CC BY-SA 3.0
2017-09-26T10:34:45.457Z
This is not a duplicate. In the linked to problem, marginal productivity is not (strictly) decreasing.
Michael Greinecker
<microeconomics><production-function>
Example production function with increasing returns to scale but diminishing marginal product
Alice
# Example production function with increasing returns to scale but diminishing marginal product

(Asked by: username_0 on 2017-09-24)

I know that diminishing marginal returns even to all factors of production doesn't imply decreasing returns to scale. But could you please give me just an example of such production function?

**Tags:** microeconomics, production-function

> **Comments:**
> **username_1**: In the title you write "increasing", in the question you write "decreasing". Correct and clarify.  
> **username_2**: This is not a duplicate. In the linked to problem, marginal productivity is not (strictly) decreasing.  

---

## 2 answers

### Answer by username_2 on 2017-09-24. Score: 3

Decreasing marginal returns to a factor means that keeping the other factors fixed, the marginal output generated by this factor is decreasing. When looking at returns to scale, we change all outputs. Increasing a factor with decreasing marginal returns can have an indirect effect in increasing the marginal productivity of other factors. If we increase all factors at the same time, the indirect effects may outweigh the direct effect. The production function $F:\mathbb{R}\_+^2$ given by
$$F(x,y)=(x+1)^{2/3}(y+1)^{2/3}$$
has decreasing marginal factor productivities everywhere but not decreasing returns to scale (it doesn't have increasing returns to scale either).


---

### Answer by username_1 on 2017-09-24. Score: 2

For decreasing marginal returns we require second partial derivatives to be negative, since we examine what happens if we vary only one input

So any function
$$y = \prod\_{i=1}^{m}x\_i^{a\_i},\;\;\; 0<a\_i<1\;\; \forall \,i$$

with $\sum a\_i >1$ for increasing returns to scale, and with $\sum a\_i <1$ for decreasing returns to scale, since here we examine what happens of we increase by the same proportion all inputs.

---

**Responding to comments**

We check returns to scale for this function by examining, for $k>1$, the expression

$$ \prod\_{i=1}^{m}(kx\_i)^{a\_i} = k^{\sum a\_i}\cdot y$$

If the sum of the alphas is higher than unity, output increases more than $k$ so we have increasing returns to scale, and correspondingly for deceasing returns to scale when the sum of the alphas is smaller than unity.

Regarding decreasing marginal returns, the rate of change of marginal output generated by a factor keeping the others fixed is given by its own second partial derivative, which here is

$$\frac {\partial^2 y}{\partial x\_i^2}= a\_i(a\_i-1)\cdot \frac {y}{x\_i^2}$$

When $0<a\_i<1$, these second partials are all negative.

> **Comments:**
> **username_2**: What is your definition of decreasing/increasing returns to scale? For example, under the [definition](https://en.wikipedia.org/wiki/Returns_to_scale#Formal_definitions) used in the Wikipedia entry on returns to scale, a Cobb-Douglas production function cannot have positive or decreasing returns to scale (something the entry later gets wrong).  
> **username_2**: Moreover, this function doesn't have decreasing marginal returns everywhere, unless $m=1$.  
> **username_1**: @username_2 I expanded my answer to answer your questions.  
> **username_2**: You are ignoring the boundary, where some factors can be zero. In that case, the whole output is zero and remains so after doubling, and the marginal productivity of zero inputs is zero and therefore constant.  
> **username_1**: @username_2 I certainly do, since it is a mathematical case which is rather uninteresting from the point of view of Economics. As regards the definition in wikipedia for the Cobb-Douglas function, it follows the convention to call "Cobb-Douglas" the function when the sum of exponents equals exactly unity, while when we want to use the exponential form with sum of exponents different than unity we usually call it "generalized Cobb-Douglas".  
> **username_2**: I would like to see your definition of returns to scale that, unlike the definition used by the Wikipedia entry, exclude "mathematical case which is rather uninteresting from the point of view of Economics."  
> **username_1**: @username_2 I am not sure that I understand what more should I write, except perhaps by adding the qualifier $x_i>0,\, \forall\, i$. All Economics textbooks I know don't even bother to mention that, exactly because a "zero-output case due to fixing an input to zero" is uninteresting for the purposes of Economics.  
> **username_2**: I do hold the view that all economic textbooks can be wrong, in particular, those that make claims to mathematical rigor. A textbook that does not commit the mistake is "Microeconomic Foundations I" by David Kreps.  

---

