# Can we characterize the set of neoclassical production functions?

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 $$F(\lambda K,\lambda L, A)=\lambda F(K,L,A).$$

$$F$$ 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.

• Do you want $\mathbb{R}^+$ to be the set of (strictly) positive numbers instead? $K^{\alpha}$ is not differentiable at 0 for $\alpha \in (0,1)$. – Kevin Casto Feb 1 at 19:20
• @KevinCasto (+1) Thanks for your comment. No. Assumption 2 requires that $F(0,K,L)$ is defined. However, I want to make the restriction that the neoclassical production function is only differentiable over open sets in its domain. The Cobb-Douglas production function should satisfy this requirement. I revised the description of Assumption 1 to account for your example. My previous description of Assumption 1 was taken directly from page 29 in Acemoglu (2009). – Elias Feb 1 at 19:36
• That's what's called a "one sector" model. You occasionally see two-sector models, etc. The main reason why you see one-sector Cobb-Douglas models is that developed economies seem to satisfy the "balanced growth" condition, and it's easy to get that condition out of one-sector Cobb-Douglas. This blog post gives the condition, and talks other kinds of models that can satisfy it: growthecon.com/blog/BGP – arsmath Feb 1 at 22:03
• @arsmath (+1) Thanks for your comment. I agree that one reason for why we often see the Cobb-Douglas production function in textbooks is that it is consistent with Uzawa's theorem and with the empirically observed constant growth rate regime during the period after the industrial revolution in countries that are developed today. However, Uzawa's theorem does not imply a Cobb-Douglas production function. It implies a Harrod-neutral production function, and the Cobb-Dogulas production function is one example. See Theorem 2.6 on page 60 in Acemoglu (2009). The link does not answer my questions. – Elias Feb 1 at 22:08
• That's why I left a comment. I wanted to draw your attention to the two-sector models mentioned in that blog post. – arsmath Feb 2 at 9:33

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.
• 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.