# Does a neoclassical production with constant returns to scale implies type of Cobb-Douglas

Assume the neoclassical production function $$F(K,L)\colon [0,\infty) \times [0,\infty) \to [0,\infty)$$ twice continuously differentiable, i.e., F is montone increasing and concave, i.e., $$\partial F(\bar K,\bar L) /\partial K > 0, \quad \partial F(\bar K,\bar L) /\partial L > 0,\\ \partial^2 F(\bar K,\bar L) /\partial K^2 < 0, \quad \partial^2 F(\bar K,\bar L) /\partial L^2 < 0,$$holds, i.e., $$F$$ has diminishing returns, for all $$\bar K, \bar L \in (0,\infty)$$. Moreover, the Inada conditions hold, i.e., $$\lim_{K\to 0} \partial F(\bar K,\bar L)/\partial K = \infty, \quad \lim_{K\to \infty} \partial F(\bar K,\bar L)/\partial K = 0,\\ \lim_{L\to 0} \partial F(\bar K,\bar L)/\partial L = \infty, \quad \lim_{L\to \infty} \partial F(\bar K,\bar L)/\partial L = 0$$ for all $$\bar K, \bar L \in (0,\infty)$$. Finally $$F$$ yields constant returns to scale, i.e., $$F$$ is positive homogeneous of degree $$1$$, i.e., $$F(\lambda K, \lambda L) = \lambda F(K,L)$$ for all $$\lambda \in (0,\infty)$$ and $$K,L \in [0,\infty)$$.

I am aware of the paper "Inada conditions imply that production function must be asymptotically Cobb–Douglas" , by Barelli and Pessôa.

So the Inada conditions with the monotonicity and concavity imply asymptotically Cobb-Douglas behaviour of every $$F$$ fulfilling those conditions. However, this argument is without imposing a constant returns to scale assumption.

The question is: Does diminishing returns + Inada conditions + constant returns to scale uniquely determine $$F$$ as given by $$F(K,L) = c K^\alpha L^{1-\alpha}$$ with $$c \in (0,\infty)$$ and $$\alpha \in (0,1)$$?

I know without constant returns to scale in 1D, i.e. utility functions twice continously differentiable that are monotone increasing and concave and fulfill Inada conditions, we can choose two functions, e.g., $$\tilde u(c)$$ and $$\bar u(c)$$, with $$\tilde u \neq \bar u$$, where $$\tilde u(c_0) = \bar u(c_0)$$, $$\tilde u'(c_0) = \bar u'(c_0)$$, and $$\tilde u''(c_0) = \bar u''(c_0)$$ holds. Then is the piecewise defined function $$u(c) = \begin{cases}\tilde u(c), & c \le c_0 \\ \bar u(c) & c > c_0\end{cases}$$ also monotone increasing and concave and fulfills the Inada conditions (and clearly twice continously differentiable).

Somehow i feel that the constant returns to scale enforces that this piecewise "trick" is not working. But i couldn't manage to find information about if these neoclassical conditions on the production function imply Cobb-Douglas type.

• The result on the production function being asymptotically Cobb-Douglas is not correct. Jun 23, 2022 at 16:26
• To clarify the comment for others, reading the content without following the links, the result that the function being asymptotically Cobb-Douglas is still right but the arguments of the paper cited in the question are wrong, this is what you want to say? Jun 23, 2022 at 16:56
• No, the function need not be Cobb-Douglas. It just needs to have an elasticity of substitution asymptotically to one. The paper gives an example of that that is not CD. Jun 23, 2022 at 18:36
• I am not sure about this fact, are you aware of this paper? Jun 23, 2022 at 18:42
• I am not; I will take a look at it. Jun 23, 2022 at 18:45

$$F(K, L) = K^{\frac{1}{4}}L^\frac{3}{4}+K^{\frac{3}{4}}L^\frac{1}{4}$$

• How do we know this is not just another form of $$\sqrt{5} \cdot K^{1/3} \cdot L^{2/3} \ \ ??$$ Jun 23, 2022 at 14:32
• It cannot be of this form because your expression does not satisfy $F(K, L) = F(L, K)$ whereas mine does.
– Amit
Jun 23, 2022 at 14:35
• You're totally right, i overseen this clear fact that the sum of two Cobb-Douglas functions (with CRS) again fulfill the in the question mentioned conditions. Based on this do you know if all functions that are neoclassical in the sense of the question are given as an infinite sum or finite sum of Cobb-Douglas functions (with CRS)? I am strongly interested in how the family of functions has to look in order to fulfill the conditions to be a neoclassical production function, i.e., does there exist an "explicit" presentation of function family. Jun 23, 2022 at 14:43
• @Giskard Likewise, we can rule out all functions such that $\alpha \neq \frac{1}{2}$. So, there is only one possibility left which is $G(K, L) = 2K^{\frac{1}{2}}L^{\frac{1}{2}}$. This can be ruled out because $G(16, 1) \neq F(16, 1)$.
– Amit
Jun 23, 2022 at 14:44
• @maximilian43 For example, if we don't assume differentiability, and just want a function that satisfy CRS, diminishing returns, inada conditions then the following function also works: $F(K, L) = \min(K^{\frac{1}{4}}L^{\frac{3}{4}}, K^{\frac{3}{4}}L^{\frac{1}{4}})$
– Amit
Jun 23, 2022 at 17:43