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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.

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    $\begingroup$ Hint - you want to look at the derivatives. $\endgroup$ Commented Sep 7, 2022 at 16:34
  • $\begingroup$ Hi, could you please explain? Thanks!! $\endgroup$
    – student
    Commented Sep 7, 2022 at 16:45
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    $\begingroup$ 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? $\endgroup$
    – BrsG
    Commented Sep 7, 2022 at 16:51
  • $\begingroup$ Hi, I am looking for a pragmatic interpretation, so i could understand the concept in a better way. Thanks! $\endgroup$
    – student
    Commented Sep 8, 2022 at 14:11

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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.

Very short interpretation, if I have a production line with 10 employees and add an equipment the marginal productivity of the employees increases because they can now use this other tool, on the other hand, the marginal productivity of the other capital equipments is decreased by this because each of them is used less.

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  • $\begingroup$ Hi! Thanks a lot for your answer. Regarding the last paragraph: How does the marginal productivity of the employees improve, when there is a new equipment, if the number of employees stay the same? Wouldn't the productivity stay the same, even if there are a 1000 more machines to possibly work with? So why doesn't the increase of L lead to a decrease of MPK, or at least an unchanging MPK? $\endgroup$
    – student
    Commented Sep 9, 2022 at 10:52
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    $\begingroup$ 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 $\endgroup$ Commented Sep 9, 2022 at 14:01
  • $\begingroup$ Hi! Thanks a lot for your answer!! $\endgroup$
    – student
    Commented Sep 11, 2022 at 6:30
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    $\begingroup$ If that is all you were looking for please accept it. If that helped you but did not solve your problem please upvote! $\endgroup$ Commented Sep 12, 2022 at 11:02

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