**INTRODUCTION**


The *neoclassical [production function](https://en.wikipedia.org/wiki/Production_function)* is the main building block in [neoclassical growth theory](https://www.investopedia.com/terms/n/neoclassical-growth-theory.asp), and consequently the main building block of modern macroeconomic theory. Mathematically, the neoclassical production function $F$ is a function that satisfies Assumption 1 and 2 below. In economics, the neoclassical production function is a production function that satisfies Assumption 1 and 2 below *and* that defines the output $Y$ by a given amount of capital $K$, labor $L$, and technology $A$: $$Y=F(K,L,A).$$ 


My questions given below are only about the mathematical definition of a neoclassical production function, leaving philosophical and empirical issues aside. Assumption 1 and 2 below are taken from pages 29 and 33 in the popular graduate-level textbook in economics by Acemoglu (2009).


Typically, only the constant returns to scale (CRS) [Cobb-Douglas production function](https://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function) is given as an example of a neoclassical production function. It can be symbolically defined by $$F(K,L,A)=K^{\alpha}(AL)^{1-\alpha}$$ for some $\alpha\in(0,1)$. (This is the so-called Harrod-neutral representation of the Cobb-Douglas function and is related to [Uzawa's theorem](https://en.wikipedia.org/wiki/Uzawa%27s_theorem). There are two other equivalent representations.) In all graduate-level textbooks in macroeconomics that I know of, no other type of neoclassical production functions is defined! With less restrictive assumptions on the production function than Assumption 1 and 2, one can show that the Inada conditions imply that the production function must be asymptotically Cobb-Douglas (Barrelli et al., 2003; Litina et al., 2008).


**QUESTIONS**


Define the set of neoclassical production functions as the set of functions that satisfy Assumption 1 and 2 below. 


* Can we characterize the set of neoclassical production functions?
* What are some examples of neoclassical production functions that do
not fall in the class of Cobb-Douglas production functions?


**ASSUMPTION 1 (Continuity, Differentiability, Positive and Diminishing Marginal Products, and Constant Returns to Scale)**


The production function $F:\mathbb{R}\_+^3\to\mathbb{R}\_+$, where $\mathbb{R}\_+$ is the set of nonnegative real numbers, is twice differentiable in $K$ and $L$, and satisfies over $(K,L,A)\in\mathbb{R}^2\_{++}\times\mathbb{R}\_+$, where $\mathbb{R}\_{++}$ is the set of positive real numbers, $$F\_K(K,L,A)\equiv\frac{\partial F(K,L,A)}{\partial K}>0,\quad F\_L(K,L,A)\equiv\frac{\partial F(K,L,A)}{\partial L}>0,$$ and $$F\_{KK}(K,L,A)\equiv\frac{\partial^2 F(K,L,A)}{\partial K^2}<0,\quad F\_{LL}(K,L,A)\equiv\frac{\partial^2 F(K,L,A)}{\partial L^2}<0.$$ Moreover, $F$ exhibits constant returns to scale in $K$ and $L$. That is, $F(K,L,A)$ is homogenous of degree 1 in $(K,L)$ so that for any $\lambda\in\mathbb{R}\_+$ and $(K,L,A)\in\mathbb{R}^3\_+$ we have that $$F(\lambda K,\lambda L, A)=\lambda F(K,L,A).$$


**ASSUMPTION 2 (Inada Conditions)**


$F$ satisfies the Inada conditions $$\lim\_{K\to0^+}F\_K(K,L,A)=\infty\text{ and }\lim\_{K\to\infty}F\_K(K,L,A)=0\text{ for all }L>0\text{ and all }A\geq0,$$ and $$\lim\_{L\to0^+}F\_L(K,L,A)=\infty\text{ and }\lim\_{L\to\infty}F\_L(K,L,A)=0\text{ for all }K>0\text{ and all }A\geq0.$$ Moreover, $$F(0,L,A)=0$$ for all $(L,A)\in\mathbb{R}^2\_+$. 


**REFERENCES**


Acemoglu, D. (2009). *Introduction to Modern Economic Growth*. Princeton University Press.


Barelli, Paulo; Pessôa, Samuel de Abreu (2003). "Inada conditions imply that production function must be asymptotically Cobb–Douglas". Economics Letters. 81 (3): 361–363.


Litina, Anastasia; Palivos, Theodore (2008). "Do Inada conditions imply that production function must be asymptotically Cobb–Douglas? A comment". Economics Letters. 99 (3): 498–499. 



Can we characterize the set of neoclassical production functions?
Concerning your Question 1: "Can we characterize the set of neoclassical production functions?" -- This set is already characterized, tautologically, by its definition, as the set of all functions $F$ satisfying Assumptions 1 and 2. There seems to be no reason/way for there to exist a better characterization. 


Concerning your Question 2: "What are some examples of neoclassical production functions that do not fall in the class of Cobb-Douglas production functions?" 


* First here, note that Assumptions 1 and 2 impose almost no restrictions on the dependence of $F(K,L,A)$ on $A$. So, e.g., any function $F$ given by the formula 
$$F(K,L,A)=K^a L^{1-a}h(A),$$
where $a\in(0,1)$ and $h$ is any positive function, will do.
* Second, note that your set of neoclassical production functions is a cone. So, any linear combination of functions in this set with positive coefficients, as well as appropriate limits of such linear combinations, will do. E.g., all the functions $F$ given by the formula 
$$F(K,L,A)=Lf(K/L)h(A)\tag{1}$$
will do, where $h$ is any positive function and 
$$f(z)=\int\_{(0,1)}z^a\mu(da)$$
for all all real $z\ge0$, where in turn $\mu$ is any positive measure on $(0,1)$. In particular, here we can take 
$$f(z)=\int\_{(0,1)}z^a\,da=\frac{z-1}{\ln z}$$
for $z\in(0,1)\cup(1,\infty)$, with $f(0)$ and $f(1)$ defined by continuity, 
$$f(z)=\int\_{(0,1)}z^a\,a\,da=\frac{1-z+z \ln z}{\ln^2 z}$$
for $z\in(0,1)\cup(1,\infty)$, with $f(0)$ and $f(1)$ defined by continuity, or, more generally,
$$f(z)=\int\_{(0,1)}z^a\,a^n\,da= \frac{\Gamma (n+1)-\Gamma (n+1,-\ln z)}
{(-\ln z)^{n+1}}$$
for real $n\ge0$ and $z\in(0,1)\cup(1,\infty)$, with $f(0)$ and $f(1)$ defined by continuity, where $\Gamma (m,u)$ represents the incomplete Gamma function. 
Of course, here one can also use (possibly infinite) linear combinations with different functions $h$.
* Third, one can certainly construct neoclassical production functions $F$ as in (1) with $f$ other than (possibly infinite) linear combinations of the functions $z\mapsto z^a$ with $a\in(0,1)$ (as was done in the second item above). For instance, one can glue pieces of such linear combinations over non-overlapping subintervals of $[0,\infty)$ in any manner just so that to preserve the continuity and differentiability.


