<p>I am assuming that you are interested in finding the aggregate production function when you have two plants and they use same inputs. So if you have <span class="math-container">$k$</span> units of capital and <span class="math-container">$l$</span> units of labor in total, how to allocate these in two plants to get the aggregate production function as a function of <span class="math-container">$k$</span> and <span class="math-container">$l$</span>. In this problem, you can first find the aggregate production function by solving this problem:</p>
<p><span class="math-container">\begin{eqnarray*} \max_{(k_1, k_2, l_1, l_2) \in \mathbb{R}^4_+} &amp; Ak_1^\alpha l_1^{1-\alpha} + Ak_2^\alpha l_2^{1-\alpha} \\ \text{s.t} \ &amp; k_1+ k_2 = k \\ \text{and} \ &amp; l_1+ l_2 = l\end{eqnarray*}</span>
and you'll get the aggregate production function <span class="math-container">$f(k, l)$</span> which is the optimal value of the objective function in the above problem.
Solving the above problem, we'll get <span class="math-container">$k_1 = k_2 = \frac{k}{2}$</span> and <span class="math-container">$l_1 = l_2 = \frac{l}{2}$</span> as one of the solutions and therefore, <span class="math-container">$f(k, l) = Ak^\alpha l^{1-\alpha}$</span> which satisfy CRS.</p>

CC BY-SA 4.0
2022-06-13T03:21:09.280Z
Thank you @Amit sir! What if $$k_1≠k_2 and l_1≠l_2$$ ? How will we check for Returns to scale then? Thanks a ton
Elina Gilbert
CC BY-SA 4.0
2022-06-13T03:40:05.800Z
It is one of the solutions to the above problem. There are other solutions and of course they'll all result in the same aggregate production function. For example: $k_1 = l_1 = 0$ and $k_2 = k, l_2 = l$ also solves the maximisation problem.
Amit
CC BY-SA 4.0
2022-06-13T04:39:06.807Z
My question is not about solution or does checking returns to scale depends on solution? Does the aggregate function, in its current form, exhibit CRS?
Elina Gilbert
CC BY-SA 4.0
2022-06-13T04:43:33.123Z
What you have written is not the aggregate production function. Anyways, if we think of $k_1$ and $k_2$ as different types of capital and $l_1$ and $l_2$ as different types of labor i.e. there are four inputs. In that case, we can write the aggregate production function as $F(l_1, k_1, l_2, k_2) = Ak_1^\alpha l_1^{1-\alpha} + Ak_2^\alpha l_2^{1-\alpha}$, and this function also clearly exhibits CRS.
Amit
CC BY-SA 4.0
2022-06-13T05:37:00.413Z
Basically, $k_1$ and $k_2$ are capital given to two individuals such that $k_1 + k_2 = k$ while $l_1$ and $l_2$ are labor inputs of two individuals. They both are producing separately and Aggregate production function, given above, summates the output produced by them. So, how can check returns of scale under such Aggregate production function where $k_1$ and $k_2$ are not different/separate types of capital.
Elina Gilbert
CC BY-SA 4.0
2022-06-13T05:42:08.227Z
If your objective is to maximise the total production and want to allocate inputs to them accordingly, then you can use $f(k, l)$. If you have no such objective then you can consider the function $F(l_1, k_1, l_2, k_2)$ if you want.
Amit
CC BY-SA 4.0
2022-06-13T05:50:10.867Z
So, when I use f(k,l) then can I solve it assuming CRS because separately production function exhibits CRS? Or Do the power of inputs should be $\alpha/2$ and $(1-\alpha)/2$ for it to be CRS?
Elina Gilbert
CC BY-SA 4.0
2022-06-13T06:04:01.043Z
I haven't understood your question.
Amit
CC BY-SA 4.0
2022-06-13T06:16:09.910Z
My problem is - do we check for CRS by adding the powers of each input, like here $ 2 [\alpha + (1-\alpha)] > 1$. That will make the production function IRS. Or do we add the power of of separate production function $ [\alpha + (1-\alpha)] $ and take them out as common as in $\lambda^{\alpha + (1-\alpha)} $ which will make above function CRS.
Elina Gilbert
CC BY-SA 4.0
2022-06-13T06:28:14.103Z
Why will you add the powers like this? Just read the definition of CRS and apply it here and you'll know. It is easy.
Amit
CC BY-SA 4.0
2022-06-13T06:40:06.427Z
Because CRS requires output to double when $all$ the inputs double. Here summing over each input power with the base 2 more than doubles the output. Isn't that the correct definition of CRS?
Elina Gilbert
CC BY-SA 4.0
2022-06-13T06:55:25.900Z
Definition is correct but you are not applying it correctly.
Amit
CC BY-SA 4.0
2022-06-13T07:02:08.127Z
So, correct way is $$\lambda^{\alpha + 1- \alpha}(Ak_1^\alpha l_1^{1-\alpha} + Ak_2^\alpha l_2^{1-\alpha}) $$ which makes the production function CRS. We did this because when inputs gets doubled their individual was CRS and Aggregate would just double everything as well. Is my reasoning correct now?
Elina Gilbert
CC BY-SA 4.0
2022-06-13T07:04:20.423Z
Yes it is correct
Amit
CC BY-SA 4.0
2022-06-13T07:05:56.997Z
Thank you sir for all the help!
Elina Gilbert
CC BY-SA 4.0
2022-06-13T02:31:03.820Z
2022-06-13T02:39:44.203Z
2022-06-13T02:39:44.203Z
Amit
<p>I have following Aggregate production function -</p>
<p><span class="math-container">$$ Ak_1^\alpha l_1^{1-\alpha} +  Ak_2^\alpha l_2^{1-\alpha}$$</span></p>
<p>Individually each part exhibit constant returns to scale but as an Aggregate function function, does it also exhibit CRS. How to check for Returns to scale in Aggregate production function?</p>

<production-function><capital-returns><aggregate-production>
Aggregate production function and returns to scale
Elina Gilbert
# Aggregate production function and returns to scale

(Asked by: username_0 on 2022-06-13)

I have following Aggregate production function -

$$ Ak\_1^\alpha l\_1^{1-\alpha} + Ak\_2^\alpha l\_2^{1-\alpha}$$

Individually each part exhibit constant returns to scale but as an Aggregate function function, does it also exhibit CRS. How to check for Returns to scale in Aggregate production function?

**Tags:** production-function, capital-returns, aggregate-production


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## 1 answers

### Answer by username_1 on 2022-06-13. Score: 6 | Accepted answer

I am assuming that you are interested in finding the aggregate production function when you have two plants and they use same inputs. So if you have $k$ units of capital and $l$ units of labor in total, how to allocate these in two plants to get the aggregate production function as a function of $k$ and $l$. In this problem, you can first find the aggregate production function by solving this problem:

\begin{eqnarray\*} \max\_{(k\_1, k\_2, l\_1, l\_2) \in \mathbb{R}^4\_+} & Ak\_1^\alpha l\_1^{1-\alpha} + Ak\_2^\alpha l\_2^{1-\alpha} \\ \text{s.t} \ & k\_1+ k\_2 = k \\ \text{and} \ & l\_1+ l\_2 = l\end{eqnarray\*}
and you'll get the aggregate production function $f(k, l)$ which is the optimal value of the objective function in the above problem.
Solving the above problem, we'll get $k\_1 = k\_2 = \frac{k}{2}$ and $l\_1 = l\_2 = \frac{l}{2}$ as one of the solutions and therefore, $f(k, l) = Ak^\alpha l^{1-\alpha}$ which satisfy CRS.

> **Comments:**
> **username_0**: Thank you @username_1 sir! What if $$k_1≠k_2 and l_1≠l_2$$ ? How will we check for Returns to scale then? Thanks a ton  
> **username_1**: It is one of the solutions to the above problem. There are other solutions and of course they'll all result in the same aggregate production function. For example: $k_1 = l_1 = 0$ and $k_2 = k, l_2 = l$ also solves the maximisation problem.  
> **username_0**: My question is not about solution or does checking returns to scale depends on solution? Does the aggregate function, in its current form, exhibit CRS?  
> **username_1**: What you have written is not the aggregate production function. Anyways, if we think of $k_1$ and $k_2$ as different types of capital and $l_1$ and $l_2$ as different types of labor i.e. there are four inputs. In that case, we can write the aggregate production function as $F(l_1, k_1, l_2, k_2) = Ak_1^\alpha l_1^{1-\alpha} + Ak_2^\alpha l_2^{1-\alpha}$, and this function also clearly exhibits CRS.  
> **username_0**: Basically, $k_1$ and $k_2$ are capital given to two individuals such that $k_1 + k_2 = k$ while $l_1$ and $l_2$ are labor inputs of two individuals. They both are producing separately and Aggregate production function, given above, summates the output produced by them. So, how can check returns of scale under such Aggregate production function where $k_1$ and $k_2$ are not different/separate types of capital.  
> **username_1**: If your objective is to maximise the total production and want to allocate inputs to them accordingly, then you can use $f(k, l)$. If you have no such objective then you can consider the function $F(l_1, k_1, l_2, k_2)$ if you want.  
> **username_0**: So, when I use f(k,l) then can I solve it assuming CRS because separately production function exhibits CRS? Or Do the power of inputs should be $\alpha/2$ and $(1-\alpha)/2$ for it to be CRS?  
> **username_1**: I haven't understood your question.  
> **username_0**: My problem is - do we check for CRS by adding the powers of each input, like here $ 2 [\alpha + (1-\alpha)] > 1$. That will make the production function IRS. Or do we add the power of of separate production function $ [\alpha + (1-\alpha)] $ and take them out as common as in $\lambda^{\alpha + (1-\alpha)} $ which will make above function CRS.  
> **username_1**: Why will you add the powers like this? Just read the definition of CRS and apply it here and you'll know. It is easy.  
> **username_0**: Because CRS requires output to double when $all$ the inputs double. Here summing over each input power with the base 2 more than doubles the output. Isn't that the correct definition of CRS?  
> **username_1**: Definition is correct but you are not applying it correctly.  
> **username_0**: So, correct way is $$\lambda^{\alpha + 1- \alpha}(Ak_1^\alpha l_1^{1-\alpha} + Ak_2^\alpha l_2^{1-\alpha}) $$ which makes the production function CRS. We did this because when inputs gets doubled their individual was CRS and Aggregate would just double everything as well. Is my reasoning correct now?  
> **username_1**: Yes it is correct  
> **username_0**: Thank you sir for all the help!  

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