Neoclassical production function

Cobb-Douglas function as an example of a neoclassical production function

In neoclassical production theory, a neoclassical production function is a continuous substitutional production function that fulfills certain properties. The neoclassical production function is called a theoretical construct, the existence of which is assumed. It is not derived from empirical observations or technical contexts. Nevertheless, its properties can be proven again and again in later approaches to production theory and it serves as the basis for theories and models based on it. The neoclassical production function is derived from the classical production function in that the inefficient areas of the classical production function are replaced by the dominant course. Neoclassical production functions are mostly used in macroeconomics , more precisely in growth theory .

Requirements for a neoclassical production function

Assume that the production relationships of a company can be represented by a monotonous, twice continuously differentiable map : ${\displaystyle f}$

${\displaystyle f\colon \mathbb {R} _{+}^{n}\to \mathbb {R} _{+}}$,

where the relationship holds. This means that the output quantity depends on the quantity used for the respective input factors . Such a function located in -dimensional goods space can be analyzed by laying a horizontal section (contour line) through the mountain of income and then "projecting" this contour line onto the factor level. Below the sake of simplicity, it is assumed that only two factors of production and exist and the function therefore is in 3-dimensional space (see illustration). The general requirements for a neoclassical production function are then: ${\displaystyle x=f(r_{1},\dotsc ,r_{n})}$ ${\displaystyle x}$ ${\displaystyle (r_{1},\dotsc ,r_{n})}$ ${\displaystyle (n+1)}$ ${\displaystyle r_{1}}$ ${\displaystyle r_{2}}$

1. The so-called Inada conditions must be met, i. This means that the marginal product of every factor of production converges towards infinity, if one only lets the respective factor input tend towards zero. If, on the other hand, the respective factor input is allowed to tend towards infinity, the marginal product of the factor converges towards zero:

   ${\displaystyle \lim _{r_{1}\to 0}{\frac {\partial f(r_{1},r_{2})}{\partial r_{1}}}=\infty \quad {\textrm {und}}\quad \lim _{r_{1}\to \infty }{\frac {\partial f(r_{1},r_{2})}{\partial r_{1}}}=0\quad \forall r_{2}>0}$

   ${\displaystyle \lim _{r_{2}\to 0}{\frac {\partial f(r_{1},r_{2})}{\partial r_{2}}}=\infty \quad {\textrm {und}}\quad \lim _{r_{2}\to \infty }{\frac {\partial f(r_{1},r_{2})}{\partial r_{2}}}=0\quad \forall r_{1}>0}$

2. Constant returns to scale or degree 1 homogeneity in effective work and capital. In economic terms, this means: An increased / decreased use of these production factors leads to an increased / decreased production in the same ratio:

   ${\displaystyle f(\lambda r_{1},\lambda r_{2})=\lambda f(r_{1},r_{2})\quad \forall \lambda \in \mathbb {R} ^{+}}$

3. Positive and decreasing marginal returns: The marginal returns of capital and effective labor are positive, but decrease as the respective factor increases. For example, if more effective labor is used, production will increase, but it will rise less if much effective labor is already being used. Mathematically this means that the first partial derivatives of the production function according to effective labor and capital are positive, but the respective second derivatives are negative:

   ${\displaystyle {\frac {\partial f(r_{1},r_{2})}{\partial r_{1}}}>0,\;\;{\frac {\partial ^{2}f(r_{1},r_{2})}{\partial r_{1}^{2}}}<0}$

   ${\displaystyle {\frac {\partial f(r_{1},r_{2})}{\partial r_{2}}}>0,\;\;{\frac {\partial ^{2}f(r_{1},r_{2})}{\partial r_{2}^{2}}}<0}$

4. Essentiality: From the above assumptions in connection with the Inada conditions it also follows that every factor used is essential (also: essential ). This means that an economy in a state in which there is either no capital or no work cannot generate any output. Formally:

   ${\displaystyle f(r_{1}=0,r_{2}>0)=0\quad {\textrm {und}}\quad f(r_{1}>0,r_{2}=0)=0}$.

This implies that the function always goes through the origin.

Example of a neoclassical production function

The Cobb-Douglas function is a frequently and successfully used neoclassical production function in production theory and microeconomics . A possible production function that fulfills the assumptions presented above is the following Cobb-Douglas function:

${\displaystyle Y_{t}=F(K_{t},L_{t})=T\cdot K_{t}^{\alpha }L_{t}^{1-\alpha }}$ With ${\displaystyle \alpha \in (0,1)}$

application

The application of the neoclassical production function is diverse. For example, it is the basis of the neoclassical basic model in the Solow model .

Individual evidence

1. ↑ According to Ken-Ichi Inada , who wrote it in his 1963 article On a Two-Sector Model of Economic Growth: Comments and Generalization. (In: Review of Economic Studies . 30.2, pp. 119–127).
2. ↑ A proof can be found, for example, in Färe / Primont 2002, p. 3 f.