<p>Let's find the Marginal Product of Labor, $MP_L$:
$$
\begin{align}
MP_L \equiv \frac{\partial Y(K,L)}{\partial L} &amp;=\frac{\partial }{\partial L}\left( aK + b L^{0.5} \right) \\
&amp;= 0.5\cdot b \cdot L^{-0.5}
\end{align}
$$
Now finding $\frac{dMP_L}{dK}$ is trivial:
$$
\begin{align}
\frac{dMP_L}{dK} &amp;= \frac{d}{dK} \left( 0.5\cdot b \cdot L^{-0.5}\right) = 0
\end{align}
$$
We see that it does not depend on the capital. So a change in $MP_L$</p>

CC BY-SA 3.0
2015-06-03T19:53:21.930Z
2015-06-03T19:53:21.930Z
Konstantinos
<p>Given the function:
$y=A K^\alpha L^\beta, A,\alpha,\beta &gt;0$</p>

<p>$y=aK+bL^{0.5}, a,b&gt;0$</p>

<p>Decide whether capital and labour are substitute or supportive factors of production. How to solve this question? To answer your possible questions i want to highlight the fact that this is neither my homework assignment nor anything i need to submit, furthermore i have simply no idea how to answer this question that is why i am not able to show any attempt to solve this question. </p>

<p>EDIT: My ideas where to somehow look for the slope of MRTS curve if negative this will imply perfect substitutes if not existing then theoretically it should imply supportive factors, however the answer key states that:</p>

<p>if $\frac{\Delta MP_L}{\Delta K}&gt;0 \rightarrow$ supportive</p>

<p>if $\frac{\Delta MP_L}{\Delta K}&lt;0 \rightarrow$ substitues</p>

<p>if $\frac{A\beta K^\alpha L^{\beta-1}}{A\beta \alpha K^{\alpha -1} L^{\beta-1}}&gt;0 \rightarrow $ subsitutes.</p>

<p>And here i am utterly lost.</p>

<microeconomics><production-function><producer-theory>
Substitutive or supportive factors of production
Krowskir
# Substitutive or supportive factors of production

(Asked by: username_0 on 2015-06-01)

Given the function:
$y=A K^\alpha L^\beta, A,\alpha,\beta >0$

$y=aK+bL^{0.5}, a,b>0$

Decide whether capital and labour are substitute or supportive factors of production. How to solve this question? To answer your possible questions i want to highlight the fact that this is neither my homework assignment nor anything i need to submit, furthermore i have simply no idea how to answer this question that is why i am not able to show any attempt to solve this question.

EDIT: My ideas where to somehow look for the slope of MRTS curve if negative this will imply perfect substitutes if not existing then theoretically it should imply supportive factors, however the answer key states that:

if $\frac{\Delta MP\_L}{\Delta K}>0 \rightarrow$ supportive

if $\frac{\Delta MP\_L}{\Delta K}<0 \rightarrow$ substitues

if $\frac{A\beta K^\alpha L^{\beta-1}}{A\beta \alpha K^{\alpha -1} L^{\beta-1}}>0 \rightarrow $ subsitutes.

And here i am utterly lost.

**Tags:** microeconomics, production-function, producer-theory

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## 1 answers

### Answer by username_1 on 2015-06-03. Score: 1 | Accepted answer

Let's find the Marginal Product of Labor, $MP\_L$:
$$
\begin{align}
MP\_L \equiv \frac{\partial Y(K,L)}{\partial L} &=\frac{\partial }{\partial L}\left( aK + b L^{0.5} \right) \\
&= 0.5\cdot b \cdot L^{-0.5}
\end{align}
$$
Now finding $\frac{dMP\_L}{dK}$ is trivial:
$$
\begin{align}
\frac{dMP\_L}{dK} &= \frac{d}{dK} \left( 0.5\cdot b \cdot L^{-0.5}\right) = 0
\end{align}
$$
We see that it does not depend on the capital. So a change in $MP\_L$


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