<p>To understand what is the issue here, try dutifully to examine all possible sub-cases in the production function.</p>
<p>The production function is</p>
<p><span class="math-container">$$Q_0 = \min\{K_0, L_0^{1/2}\}$$</span>.</p>
<p>Consider cases</p>
<p><strong>A.</strong> <span class="math-container">$K_0 &lt;  L_0^{1/2}$</span></p>
<p>Here <span class="math-container">$Q_0 = K_0$</span>. Consider <span class="math-container">$Q_{\lambda }\equiv \min\{\lambda K_0, \lambda^{1/2}L_0^{1/2}\},\;\;\; \lambda&gt;1.$</span></p>
<p><strong>Subcase A1.</strong><br />
If <span class="math-container">$\lambda K_0\leq \lambda^{1/2}L_0^{1/2} \implies Q_{\lambda} = \lambda K_0 = \lambda Q_0$</span> and we have <strong>Constant</strong> Returns to Scale. This happens for <span class="math-container">$K_0 \leq \left(L_0/\lambda\right)^{1/2}$</span>.</p>
<p><strong>Subcase A2.</strong><br />
If <span class="math-container">$\lambda K_0 &gt; \lambda^{1/2}L_0^{1/2} \implies Q_{\lambda} = \lambda^{1/2}L_0^{1/2}$</span> and because we are in the case <span class="math-container">$K_0\leq L_0^{1/2}$</span> and <span class="math-container">$\lambda  &gt;1$</span> we have <span class="math-container">$Q_{\lambda} &gt; Q_0$</span> and we have <strong>Increasing</strong> Reaurns to Scale. This happens for <span class="math-container">$\left(L_0/\lambda\right)^{1/2}&lt;  K_0 &lt;  L_0^{1/2}$</span>.</p>
<hr />
<p><strong>B.</strong> <span class="math-container">$K_0 &gt;  L_0^{1/2}$</span></p>
<p>Here <span class="math-container">$Q_0 = L_0^{1/2}$</span>. Consider <span class="math-container">$Q_{\lambda }\equiv \min\{\lambda K_0, \lambda^{1/2}L_0^{1/2}\},\;\;\; \lambda&gt;1.$</span></p>
<p>Here, because <span class="math-container">$\lambda &gt;1$</span>, we will <em>always</em> have  <span class="math-container">$\lambda K_0 &gt; \lambda^{1/2}L_0^{1/2} \implies Q_{\lambda} = \lambda^{1/2}L_0^{1/2} = \lambda^{1/2} Q_0$</span> and so we obtain <strong>Decreasing</strong> Returns to Scale.</p>
<p><strong>C.</strong> <span class="math-container">$K_0 =  L_0^{1/2}$</span></p>
<p>Here too, because <span class="math-container">$\lambda &gt;1$</span>, we are in the same situation as in Case B, so <strong>Decreasing</strong> returns to Scale.</p>
<hr />
<p>So it appears that the &quot;point of departure&quot; (the input situation for <span class="math-container">$Q_0$</span>) matters -and this is why in the other answer, optimal behavior was invoked, in order to pin down in some meaningful sense this &quot;point of departure&quot;.</p>

CC BY-SA 4.0
2020-12-17T17:34:32.907Z
2020-12-17T17:34:32.907Z
Alecos Papadopoulos
<p>You have a Leontief production function and in optimum you will always have <span class="math-container">$K=\sqrt{L}=q_1$</span>.  Now increase both inputs by factor <span class="math-container">$k&gt;1$</span> and you arrive at <span class="math-container">$k*K &gt; \sqrt{kL}=q_2$</span> where the first inequality follows from <span class="math-container">$k&gt;\sqrt{k}$</span> for <span class="math-container">$k&gt;1$</span> and the second ineqality from the fact that only the minimum matters in your production function.
Therefore you have <span class="math-container">$F(kK,kL)= \sqrt{kL}&lt;kK=k\sqrt{L} =k*F(K,L)$</span> with any optimal input combination <span class="math-container">$(K,L)$</span> and therefore decreasing returns to scale.</p>

CC BY-SA 4.0
2020-12-17T12:37:10.020Z
I thought returns to scale needed to satisfy for any K and L, not just at the optimum.....
Robin311
CC BY-SA 4.0
2020-12-17T12:50:22.577Z
Here, this is not true as a comment already suggested. Take $K=1$ and $L=10$ giving output $q_1=1$, then you have CRS for low $k$, e.g. $k=2$ gives you output $q_2=2=2*q_1=2$.
Bayesian
CC BY-SA 4.0
2020-12-17T12:54:06.793Z
Also note that your cost function also relies on the optimal combination of K and L.
Bayesian
CC BY-SA 4.0
2020-12-17T13:01:11.147Z
So are you saying that a counterexample could be like, 
when K < L^(1/2), F(K,L) = K 
and if kK < k^(1/2) * L^(1/2), then  F(kK, kL) = kK = kF(K,L)  which means it's CRS in this case instead of DRS ?
Robin311
CC BY-SA 4.0
2020-12-17T13:23:27.580Z
Yes, that's what I meant. Given this example, I would be curious how your professor arrives at DRS for general K,L.
Bayesian
CC BY-SA 4.0
2020-12-17T14:17:34.787Z
aha, I understand your answer now. Thanks for the explanation :)
Robin311
CC BY-SA 4.0
2020-12-17T12:04:42.447Z
2020-12-17T12:10:45.283Z
2020-12-17T12:10:45.283Z
Bayesian
<p>I learned that when there is decreasing returns to scale, the average cost is always increasing.</p>
<p>But the professor told us today that the other way around might not always be true. So if average cost is increasing, it might not necessarily mean that there is decreasing returns to scale.</p>
<p>The production function <span class="math-container">$q = \min \{K, L^{(1/2)}\}$</span> was one he gave us as something to think about as a possible counterexample.</p>
<p>we assumed that the input prices are 2 for capital and 1 for labor.</p>
<p>If we solve for this using cost minimization, we get <span class="math-container">$C(q) = 2q + q^2$</span> and the <span class="math-container">$AC(q) = 2 + q$</span> which is an increasing function.</p>
<p>And if I input <span class="math-container">$kK, kL$</span> in the production function where <span class="math-container">$k &gt; 1$</span> , then the function becomes</p>
<p><span class="math-container">$q = \min \{kK, (kL)^{1/2}\} = \min \{kK, k^{1/2} * L^{1/2}\}.$</span></p>
<p>How do I know what returns to scale this is showing?</p>
<p>if the min function was simply <span class="math-container">$q = \min \{ K, L \}$</span>, then I could definitely tell that this is showing constant returns to scale since <span class="math-container">$\min \{kK, kL\} = k*\min\{K,L\}$</span>,  but how do you mathematically solve for returns to scale for a function like the one above, <span class="math-container">$q = \min \{K, L^{1/2}\}$</span> ?</p>
<p>I don't know what step to take next after arriving at <span class="math-container">$\min \{kK, k^{1/2} * L^{1/2}\}$</span>. I don't know what to extract out of the min function from this point on.</p>
<p>I would appreciate advice on how to continue to determine the returns to scale for this kind of example.</p>

CC BY-SA 4.0
2020-12-17T09:53:18.343Z
How did you define constant/increasing/decreasing returns to scale in your course?
Michael Greinecker
CC BY-SA 4.0
2020-12-17T10:19:19.603Z
https://en.wikipedia.org/wiki/Returns_to_scale  

just like the wikipedia (Formal definitions part )

so I need to find the relationship between f(kK, kL) and kf(K,L), and the function of interest is a bit difficult for me to derive it...
Robin311
CC BY-SA 4.0
2020-12-17T10:25:47.667Z
If this has to hold for all $K$ and $L$, then neither of these categories apply.
Michael Greinecker
CC BY-SA 4.0
2020-12-17T11:19:55.607Z
could you explain what you mean in detail? Do you mean that it's not IRS, DRS, or CRS? But shouldn't a production function show one of the three for any time you multiply k ?
Robin311
CC BY-SA 4.0
2020-12-17T14:17:24.270Z
For example, increasing returns to scale means that for all $K$, all $L$, and all $\lambda>0$ one has $f(\lambda K,\lambda L)>\lambda f(K,L)$. This is violated as soon as there exists at least one $K$, $L$, and $\lambda>0$ such that the inequality holds. But that does not guarantee that the converse inequality or equality hold in every case.
Michael Greinecker
<microeconomics><production-function><cost-functions><leontief>
What is the returns to scale of the production function q = min {K, L^(1/2)}?
Robin311
# What is the returns to scale of the production function q = min {K, L^(1/2)}?

(Asked by: username_0 on 2020-12-17)

I learned that when there is decreasing returns to scale, the average cost is always increasing.

But the professor told us today that the other way around might not always be true. So if average cost is increasing, it might not necessarily mean that there is decreasing returns to scale.

The production function $q = \min \{K, L^{(1/2)}\}$ was one he gave us as something to think about as a possible counterexample.

we assumed that the input prices are 2 for capital and 1 for labor.

If we solve for this using cost minimization, we get $C(q) = 2q + q^2$ and the $AC(q) = 2 + q$ which is an increasing function.

And if I input $kK, kL$ in the production function where $k > 1$ , then the function becomes

$q = \min \{kK, (kL)^{1/2}\} = \min \{kK, k^{1/2} \* L^{1/2}\}.$

How do I know what returns to scale this is showing?

if the min function was simply $q = \min \{ K, L \}$, then I could definitely tell that this is showing constant returns to scale since $\min \{kK, kL\} = k\*\min\{K,L\}$, but how do you mathematically solve for returns to scale for a function like the one above, $q = \min \{K, L^{1/2}\}$ ?

I don't know what step to take next after arriving at $\min \{kK, k^{1/2} \* L^{1/2}\}$. I don't know what to extract out of the min function from this point on.

I would appreciate advice on how to continue to determine the returns to scale for this kind of example.

**Tags:** microeconomics, production-function, cost-functions, leontief
> **Comments:**
> **username_1**: How did you define constant/increasing/decreasing returns to scale in your course?  
> **username_0**: https://en.wikipedia.org/wiki/Returns_to_scale    just like the wikipedia (Formal definitions part )  so I need to find the relationship between f(kK, kL) and kf(K,L), and the function of interest is a bit difficult for me to derive it...  
> **username_1**: If this has to hold for all $K$ and $L$, then neither of these categories apply.  
> **username_0**: could you explain what you mean in detail? Do you mean that it's not IRS, DRS, or CRS? But shouldn't a production function show one of the three for any time you multiply k ?  
> **username_1**: For example, increasing returns to scale means that for all $K$, all $L$, and all $\lambda>0$ one has $f(\lambda K,\lambda L)>\lambda f(K,L)$. This is violated as soon as there exists at least one $K$, $L$, and $\lambda>0$ such that the inequality holds. But that does not guarantee that the converse inequality or equality hold in every case.  

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## 2 answers

### Answer by username_3 on 2020-12-17. Score: 4 | Accepted answer

You have a Leontief production function and in optimum you will always have $K=\sqrt{L}=q\_1$. Now increase both inputs by factor $k>1$ and you arrive at $k\*K > \sqrt{kL}=q\_2$ where the first inequality follows from $k>\sqrt{k}$ for $k>1$ and the second ineqality from the fact that only the minimum matters in your production function.
Therefore you have $F(kK,kL)= \sqrt{kL} with any optimal input combination $(K,L)$ and therefore decreasing returns to scale.

> **Comments:**
> **username_0**: I thought returns to scale needed to satisfy for any K and L, not just at the optimum.....  
> **username_3**: Here, this is not true as a comment already suggested. Take $K=1$ and $L=10$ giving output $q_1=1$, then you have CRS for low $k$, e.g. $k=2$ gives you output $q_2=2=2*q_1=2$.  
> **username_3**: Also note that your cost function also relies on the optimal combination of K and L.  
> **username_0**: So are you saying that a counterexample could be like,  when K < L^(1/2), F(K,L) = K  and if kK < k^(1/2) * L^(1/2), then  F(kK, kL) = kK = kF(K,L)  which means it's CRS in this case instead of DRS ?  
> **username_3**: Yes, that's what I meant. Given this example, I would be curious how your professor arrives at DRS for general K,L.  
> **username_0**: aha, I understand your answer now. Thanks for the explanation :)  

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### Answer by username_2 on 2020-12-17. Score: 4

To understand what is the issue here, try dutifully to examine all possible sub-cases in the production function.

The production function is

$$Q\_0 = \min\{K\_0, L\_0^{1/2}\}$$.

Consider cases

**A.** $K\_0 < L\_0^{1/2}$

Here $Q\_0 = K\_0$. Consider $Q\_{\lambda }\equiv \min\{\lambda K\_0, \lambda^{1/2}L\_0^{1/2}\},\;\;\; \lambda>1.$

**Subcase A1.**  
If $\lambda K\_0\leq \lambda^{1/2}L\_0^{1/2} \implies Q\_{\lambda} = \lambda K\_0 = \lambda Q\_0$ and we have **Constant** Returns to Scale. This happens for $K\_0 \leq \left(L\_0/\lambda\right)^{1/2}$.

**Subcase A2.**  
If $\lambda K\_0 > \lambda^{1/2}L\_0^{1/2} \implies Q\_{\lambda} = \lambda^{1/2}L\_0^{1/2}$ and because we are in the case $K\_0\leq L\_0^{1/2}$ and $\lambda >1$ we have $Q\_{\lambda} > Q\_0$ and we have **Increasing** Reaurns to Scale. This happens for $\left(L\_0/\lambda\right)^{1/2}< K\_0 < L\_0^{1/2}$.

---

**B.** $K\_0 > L\_0^{1/2}$

Here $Q\_0 = L\_0^{1/2}$. Consider $Q\_{\lambda }\equiv \min\{\lambda K\_0, \lambda^{1/2}L\_0^{1/2}\},\;\;\; \lambda>1.$

Here, because $\lambda >1$, we will *always* have $\lambda K\_0 > \lambda^{1/2}L\_0^{1/2} \implies Q\_{\lambda} = \lambda^{1/2}L\_0^{1/2} = \lambda^{1/2} Q\_0$ and so we obtain **Decreasing** Returns to Scale.

**C.** $K\_0 = L\_0^{1/2}$

Here too, because $\lambda >1$, we are in the same situation as in Case B, so **Decreasing** returns to Scale.

---

So it appears that the "point of departure" (the input situation for $Q\_0$) matters -and this is why in the other answer, optimal behavior was invoked, in order to pin down in some meaningful sense this "point of departure".


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