# How do you determine if the production function has decreasing returns to scale?

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?

• In order to answer this question you need to derive the second order conditions with respect both to k and l – Yorgos Jan 15 '17 at 18:49
• 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 Jan 15 '17 at 18:52
• 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 Jan 15 '17 at 18:58
• 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 Jan 15 '17 at 19:00
• you can have a look at economics.stackexchange.com/questions/13349/… – Yorgos Jan 15 '17 at 19:31

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 $ε_l+ε_k< 1$

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.

• How is this different from the current answer? – Giskard Nov 14 '18 at 19:26