# In the Cobb-Douglas production function, why do K and L have these effects?

Why does an increase in K lead to an increase of MPL, and to a decrease of MPK?

Likewise, why does an increase in L lead to a decrease of MPL, and to an increase of MPK?

Thank you.

• Hint - you want to look at the derivatives. Sep 7, 2022 at 16:34
• Hi, could you please explain? Thanks!! Sep 7, 2022 at 16:45
• It's a property of the Cobb-Douglas function. So all you need to do is to investigate it, that is, look at the derivatives, as RegressForward suggests! Or are you after an interpretation?
– BrsG
Sep 7, 2022 at 16:51
• Hi, I am looking for a pragmatic interpretation, so i could understand the concept in a better way. Thanks! Sep 8, 2022 at 14:11

Take a CD production function: $$Y=K^\alpha L^\beta$$ and define MP of input I as $$MP_i = \frac{\partial Y}{\partial i}$$ where i is K,L.
It is immediate to see that $$MPL = \beta K^\alpha L^{\beta-1}$$ $$MPK = \alpha K^{\alpha-1} L^\beta$$
Now you can find the derivatives $$\frac{\partial MPL}{\partial K}=\alpha\beta K^{\alpha-1}L^{\beta-1}$$,$$\frac{\partial MPK}{\partial K}=\alpha(\alpha-1)K^{\alpha-2}L^\beta$$ (you should try that with MPL). Now with $$\alpha\leq1$$ that is the case if you impose standard CRS ($$\alpha+\beta=1$$) you have that the cross derivative is always positive, while the second is negative or zero.
• That was just an example and real world may be different. However, for the effect of the $\frac{\partial MPL}{\partial K}$ think about an office, if you have 10 employees and 5 computers an increase in the number of pc will raise worker's marginal productivity (up to a certain point). As with $\frac{\partial MPK}{\partial L}$ reverse the situation 5 empl. and 10 pcs... BTW you'd find all of this and more in any intermediate micro textbook Sep 9, 2022 at 14:01