<p>They are two different concepts I think. </p>

<p>Diminishing product means that holding other things fixed, one unit of extra input($K$ or $L$ here) yields less and less additional output, which you have known. </p>

<p>Meanwhile DRS is saying that if you multiply both $K$ and $L$ by some scalar $ t &gt; 1$, the corresponding output would be less than $t$ times the original output. </p>

CC BY-SA 3.0
2016-10-30T20:33:45.767Z
so then I simply assign any +ive value $\forall{L,K}$ and it would demonstrate such?
Bensstats
CC BY-SA 3.0
2016-10-30T20:40:50.127Z
@PloniAlmoni $\forall t>1$, $F(tK, tL) = (tK)^{1/4}(tL)^{1/4} = t^{1/4+1/4}F(K, L)$. So you can see it does exhibit DRS here. In fact, in this form of production function you can easily see that $L^\alpha K^\beta$ will be DRS whenever $\alpha+\beta<1$.
Eric Chen
CC BY-SA 3.0
2016-10-30T20:31:58.567Z
2016-10-30T20:31:58.567Z
Eric Chen
<h1>The Question</h1>
<p>Suppose a firm has a production function given by</p>
<p><span class="math-container">$$y=F(L,K)=L^{1/4}K^{1/4}$$</span></p>
<p>where L and K denote inputs used in the production of y units of output.</p>
<p><strong>(a)</strong> Determine whether marginal products are diminishing</p>
<p><strong>(b)</strong> Show that production technology exhibits decreasing returns to scale</p>
<h1>My attempt</h1>
<p><strong>(a)</strong> So the marginal products, <span class="math-container">$MP_L$</span> , <span class="math-container">$MP_K$</span> are:</p>
<p><span class="math-container">$$MP_L={\partial{F}\over\partial{L}}={1\over{4}}L^{-3\over{4}}K^{1\over{4}}$$</span></p>
<p><span class="math-container">$$MP_k={\partial{F}\over\partial{K}}={1\over{4}}L^{1\over{4}}K^{-3\over{4}}$$</span></p>
<p>To determine if the marginal products are diminishing one needs to simply derive the equations again. Which would be:</p>
<p><span class="math-container">$${\partial{MP_L}\over{\partial{L}}}={-3\over{16}}L^{-7\over{4}}K^{1/4}$$</span></p>
<p>and</p>
<p><span class="math-container">$${\partial{MP_k}\over{\partial{K}}}={-3\over{16}}L^{1\over{4}}K^{-7\over{4}}$$</span></p>
<p>When both Marginal products are derived, their results are both are <span class="math-container">$&lt;0$</span> which would imply that they are diminishing.</p>
<p><strong>(b)</strong> This is where I get a little confused, is it not because we know that the Marginal products are diminishing, we know that the production technology exhibits decreasing returns to scale?</p>

CC BY-SA 3.0
2016-10-30T20:29:19.587Z
Marginal product is about how output changes when only one input changes. Returns to scale is about how output changes when all inputs change in same proportion.
Paul
<microeconomics><mathematical-economics>
Showing that production technology exhibits decreasing returns to Scale
Bensstats
# Showing that production technology exhibits decreasing returns to Scale

(Asked by: username_0 on 2016-10-30)

# The Question

Suppose a firm has a production function given by

$$y=F(L,K)=L^{1/4}K^{1/4}$$

where L and K denote inputs used in the production of y units of output.

**(a)** Determine whether marginal products are diminishing

**(b)** Show that production technology exhibits decreasing returns to scale

# My attempt

**(a)** So the marginal products, $MP\_L$ , $MP\_K$ are:

$$MP\_L={\partial{F}\over\partial{L}}={1\over{4}}L^{-3\over{4}}K^{1\over{4}}$$

$$MP\_k={\partial{F}\over\partial{K}}={1\over{4}}L^{1\over{4}}K^{-3\over{4}}$$

To determine if the marginal products are diminishing one needs to simply derive the equations again. Which would be:

$${\partial{MP\_L}\over{\partial{L}}}={-3\over{16}}L^{-7\over{4}}K^{1/4}$$

and

$${\partial{MP\_k}\over{\partial{K}}}={-3\over{16}}L^{1\over{4}}K^{-7\over{4}}$$

When both Marginal products are derived, their results are both are $<0$ which would imply that they are diminishing.

**(b)** This is where I get a little confused, is it not because we know that the Marginal products are diminishing, we know that the production technology exhibits decreasing returns to scale?

**Tags:** microeconomics, mathematical-economics
> **Comments:**
> **username_1**: Marginal product is about how output changes when only one input changes. Returns to scale is about how output changes when all inputs change in same proportion.  

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## 1 answers

### Answer by username_2 on 2016-10-30. Score: 4 | Accepted answer

They are two different concepts I think.

Diminishing product means that holding other things fixed, one unit of extra input($K$ or $L$ here) yields less and less additional output, which you have known.

Meanwhile DRS is saying that if you multiply both $K$ and $L$ by some scalar $ t > 1$, the corresponding output would be less than $t$ times the original output.

> **Comments:**
> **username_0**: so then I simply assign any +ive value $\forall{L,K}$ and it would demonstrate such?  
> **username_2**: @PloniAlmoni $\forall t>1$, $F(tK, tL) = (tK)^{1/4}(tL)^{1/4} = t^{1/4+1/4}F(K, L)$. So you can see it does exhibit DRS here. In fact, in this form of production function you can easily see that $L^\alpha K^\beta$ will be DRS whenever $\alpha+\beta<1$.  

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