<p>Note that your production function has the functional form of a Cobb-Douglas. These types of functions have some interesting properties. One of them is that the power of each input is simply its elasticity. </p>

<p>In other words, $ε_k=1.4$ and $ε_l=0.5$.</p>

<p>In order to see whether $f$ has diminishing or increasing returns you need -as @denesp pointed to me- to see $f(tk,tl)\gtrless tf(k,l)$, where $t&gt;1$.</p>

<p>One property of a Cobb-Douglas production function is that, $f(tk,tl)=t^{ε_l+ε_k}f(k,l)$.</p>

<p>So, when you have a C-D production function you can conclude about its productivity by summing its inputs elasticities. </p>

<p>In your case, you have an increasing returns Cobb-Douglas production function if $ε_l+ε_k&gt; 1$, and you have a diminishing returns C-D production function if $ε_l+ε_k&lt; 1$ </p>

CC BY-SA 3.0
2017-01-15T22:59:06.650Z
2017-01-16T00:50:10.690Z
2017-01-16T00:50:10.690Z
Yorgos
<p><a href="https://i.sstatic.net/ApsnP.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ApsnP.png" alt="Consider a Cobb-Douglas function"></a></p>

<p>The Cobb-Douglas technology’s returns-to-scale is</p>

<pre><code>constant      if   a1+ … + an  = 1
increasing   if   a1+ … + an  &gt; 1
decreasing  if   a1+ … + an  &lt; 1.
</code></pre>

<p>In your case, a1+a2=1.4+0.5=1.9, which is greater than 1. Therefore the given function exhibits increasing returns-to-scale.</p>

CC BY-SA 4.0
2018-11-14T19:26:56.697Z
How is this different from the current answer?
Giskard
CC BY-SA 4.0
2018-11-14T17:38:51.427Z
2018-11-14T17:38:51.427Z
lord AJ
<p>How do you determine this for the production function $f(k,l) = k^{1.4}l^{0.5}$ ?</p>

<p>So far, I have found the marginal product of both labour and capital however, the marginal product of labour is diminishing but the marginal product of capital is rising. Therefore, how do I determine the overall effect? Does this function display diminishing or increasing returns to scale?</p>

CC BY-SA 3.0
2017-01-15T18:49:05.353Z
In order to answer this question you need to derive the second order conditions with respect both to k and l
Yorgos
CC BY-SA 3.0
2017-01-15T18:52:36.793Z
I've only learned first order conditions so far (lagrangian multipliers to solve multiple variable questions). Is it possible to solve it like this? @Yorgos
futureecongrad
CC BY-SA 3.0
2017-01-15T18:58:01.690Z
lagrangian multipliers are used for solving constraint optimization problems. In order to find that a function has either a minimum or a maximum point you need to check second order derivatives
Yorgos
CC BY-SA 3.0
2017-01-15T19:00:24.183Z
in your example, $f$ has diminishing marginal returns if $f_{kk}<0$, $f_{ll}<0$ and $f_{kk}f_{ll}-f_{kl}f_{lk}>0$
Yorgos
CC BY-SA 3.0
2017-01-15T19:31:55.110Z
you can have a look at https://economics.stackexchange.com/questions/13349/how-to-show-the-production-function-is-concave-in-k-and-l-but-not-strictly-so?rq=1
Yorgos
<microeconomics><production-function>
How do you determine if the production function has decreasing returns to scale?
futureecongrad
# How do you determine if the production function has decreasing returns to scale?

(Asked by: username_0 on 2017-01-15)

How do you determine this for the production function $f(k,l) = k^{1.4}l^{0.5}$ ?

So far, I have found the marginal product of both labour and capital however, the marginal product of labour is diminishing but the marginal product of capital is rising. Therefore, how do I determine the overall effect? Does this function display diminishing or increasing returns to scale?

**Tags:** microeconomics, production-function

> **Comments:**
> **username_1**: In order to answer this question you need to derive the second order conditions with respect both to k and l  
> **username_0**: I've only learned first order conditions so far (lagrangian multipliers to solve multiple variable questions). Is it possible to solve it like this? @username_1  
> **username_1**: lagrangian multipliers are used for solving constraint optimization problems. In order to find that a function has either a minimum or a maximum point you need to check second order derivatives  
> **username_1**: in your example, $f$ has diminishing marginal returns if $f_{kk}<0$, $f_{ll}<0$ and $f_{kk}f_{ll}-f_{kl}f_{lk}>0$  
> **username_1**: you can have a look at https://economics.stackexchange.com/questions/13349/how-to-show-the-production-function-is-concave-in-k-and-l-but-not-strictly-so?rq=1  

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## 2 answers

### Answer by username_1 on 2017-01-15. Score: 2

Note that your production function has the functional form of a Cobb-Douglas. These types of functions have some interesting properties. One of them is that the power of each input is simply its elasticity.

In other words, $ε\_k=1.4$ and $ε\_l=0.5$.

In order to see whether $f$ has diminishing or increasing returns you need -as @denesp pointed to me- to see $f(tk,tl)\gtrless tf(k,l)$, where $t>1$.

One property of a Cobb-Douglas production function is that, $f(tk,tl)=t^{ε\_l+ε\_k}f(k,l)$.

So, when you have a C-D production function you can conclude about its productivity by summing its inputs elasticities.

In your case, you have an increasing returns Cobb-Douglas production function if $ε\_l+ε\_k> 1$, and you have a diminishing returns C-D production function if $ε\_l+ε\_k< 1$


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### Answer by username_2 on 2018-11-14. Score: 0

[![Consider a Cobb-Douglas function](https://i.sstatic.net/ApsnP.png)](https://i.sstatic.net/ApsnP.png)

The Cobb-Douglas technology’s returns-to-scale is

```
constant      if   a1+ … + an  = 1
increasing   if   a1+ … + an  > 1
decreasing  if   a1+ … + an  < 1.

```

In your case, a1+a2=1.4+0.5=1.9, which is greater than 1. Therefore the given function exhibits increasing returns-to-scale.

> **Comments:**
> **username_3**: How is this different from the current answer?  

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