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?

  • $\begingroup$ In order to answer this question you need to derive the second order conditions with respect both to k and l $\endgroup$
    – Yorgos
    Jan 15 '17 at 18:49
  • $\begingroup$ 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 $\endgroup$ Jan 15 '17 at 18:52
  • $\begingroup$ 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 $\endgroup$
    – Yorgos
    Jan 15 '17 at 18:58
  • $\begingroup$ in your example, $f$ has diminishing marginal returns if $f_{kk}<0$, $f_{ll}<0$ and $f_{kk}f_{ll}-f_{kl}f_{lk}>0$ $\endgroup$
    – Yorgos
    Jan 15 '17 at 19:00
  • $\begingroup$ you can have a look at economics.stackexchange.com/questions/13349/… $\endgroup$
    – 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 $ε_l+ε_k< 1$


Consider a Cobb-Douglas function

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.

  • 1
    $\begingroup$ How is this different from the current answer? $\endgroup$
    – Giskard
    Nov 14 '18 at 19:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.