# Aggregate production function and returns to scale

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?

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.

• 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 Commented Jun 13, 2022 at 3:21
• 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
Commented Jun 13, 2022 at 3:40
• 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? Commented Jun 13, 2022 at 4:39
• 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
Commented Jun 13, 2022 at 4:43
• Yes it is correct
– Amit
Commented Jun 13, 2022 at 7:04