Aggregating CRS Production Functions

If thera are two firms and both of them have constant returns to scale production function. Will the aggregate/industry production function still be the sum of individual production functions. How does CRS effect the aggregate production function of industry?

• For your first question, why would one think the aggregate production function not be the sum of the individual ones in this case? Nov 3 '15 at 20:55
• I don't know. I have been given two similar functions, the only difference between them is that one is decreasing returns to scale and other is constant returns to scale and I'm supposed to explain how does that effect aggregate production function. Nov 3 '15 at 21:04
• @KitsuneCavalry Why should it? Nov 4 '15 at 8:23
• It shouldn't necessarily. Nov 4 '15 at 13:28

Consider two firms with production functions $A_1 F(k,l)$ and $A_2 F(k,l)$. Both have the same curvature (and are CRS), but one is more productive. Here we can solve the social planner's problem (why?) instead of looking at a competitive equilibrium. He will want to equalize marginal returns across the firms. As one is more efficient, it will get more inputs.

And what is the maximum? Give all of the inputs to the more productive firm. This is because of CRS (which indeed means that as you scale up one production function, it's marginal product will always be larger than the other one's).

That is, define $G(l, k)$ as the aggregate production function. It is given by

$$G(l,k) = \max_{(l_i, k_i)_{i=1,2}} A_1F(k_1,l_1) + A_2F(k_2,l_2) \text{ s.t. resource constraint}$$

The solution here is $G(l,k) = A_1F(k, l)$ if $A_1 > A_2$, and vice-versa.

Now, it won't be the case that one firm is more productive than the other one always. Consider $F_1(k_1) = A_1 k_1^\alpha$ and $F_2(k_2) = A_2 k_2^\alpha$. You will want to split up capital between the two firms such that marginal returns equalize: $k_1, k_2 : F_1'(k_1) = F_2'(k_2)$. Due to DRS, this necessarily means that both $k_1, k_2 > 0$. I leave the actual derivation of the aggregate production function as an exercise.