**INTRODUCTION**

The *neoclassical production function* is the main building block in neoclassical growth theory, 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 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. 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.