In this article, I will explain the concept of the production function. The concept of the production function is generally used in the physical and biological sciences. However, the concept of production function was largely developed and extensively used in economic literature also. The development and refinement of the concepts of production functions grew out of economics due to its importance in estimation, analysis and planning of economic growth and development. It is used in determining the extent to which factors of production can be used to increase the national income of countries. The values of production coefficients serve as the basis for determining the optimum patterns of the output. So the concept of production functions is the basis for certain theories in the functional distribution of the income.

# The Concept of the Production Function

During the early part of the 18th century, the concept of production function was a very fragmentary and unsystematic speculator of industrial production behaviour. According to Malthus and Sir Edward West (1815)

"the amount of output (product) would increase at a diminishing rate when combined doses of labor and capital were applied to a given piece of land."

Ricardo (1817) also passed some wisdom as a basis for his theory of distribution:

"... the joint return to labor and capital was governed by and was equal to the amount of product added by the last combined contribution of labor and capital while the owners of the land received as rent and it seemed all in excess of these amounts."

Von Themmen’s in 1840 discovered the marginal productivity theory by breaking down the concept of a combined dose of labour and the capital and introduced the concept of *marginal productivity theory* by saying,

"... when each of the factors was separately increased keeping all the others constant and the output increases by diminishing rates."

However, the formulation of marginal productivity theory is associated with the names of J.B Clark (1889), and P.H. Wicksteed (1894) these two economists independently made the first clear cut statement about the marginal productivity theory. Though, Leon Walras in his writings during 1896 stated about this concept. The clear statement of the marginal productivity theory did not appear until the third edition emerged in 1896. His treatments of production were based on the assumption of the fixed technical coefficients with only a brief discussion about the variable coefficients. Most often the credit goes to P.H. Wicksteed (1894) as the originator of the marginal productivity theory because he gave a spectacular treatment to this subject stating:

"If the production functions were characterized by homogeneous functions of the first degree then with each factor receiving its marginal product and the total output would be absorbed in the payments to the factors without either surplus or deficit."

From the above analysis and our commonsense, it is clear that

"Production Function is a relationship between inputs and the output."

After classical, Neo-classical gave mathematical expression to the production function. Attempts were made to comprehend the concept of production function as a theory of functional distribution of income among the cooperating factors bringing to an end to the classical theory. Neo-classical made the study of production function feasible, albeit, technical by the lucid mathematical formulation of the relationship between inputs and the output. The neo-classical approach assumes a production function of the form
\[Q = f\left( {\mathop X\nolimits_1 ,\mathop X\nolimits_2 ,...\mathop X\nolimits_n } \right)\]
Where ‘Q’ being output, X_{1}, X_{2}, ... X_{n} being factors of inputs in the production process and ‘f ’ is an unspecified function form.

Thus, in Neo-classical sense, we can say that

"Production Function is a technical relationship between inputs and the output."

The relationship is said to be technical because it has been expressed in mathematical forms so some techniques are used to study this relationship. The Neo-classical production function sets some criteria for the study of the production function. In another word, some criteria are followed for quantitative analysis of production function which has been set by Neo-classical. It has been a very important subject matter of econometric analysis.

## Properties of Neo-Classical Production Function

Neo-classical production function has some nice property which makes the study of production function feasible even for beginners. Moreover, even if it is technical it gives a good insight into the actual function of the production process. Henceforth, we will assume that there are only two factors of production, namely, Labour, denoted by L, and Capital, denoted by K. This assumption is needed for the sake of simplicity in the beginning. Now, the general form of the function is expressed as follow: \[Q = f\left( {L,K} \right)\] On the basis of this equation, the Neo-classical Production Function has following properties:- Marginal Product of Labour (MP
_{L}) and Marginal Product of Capital (MP_{K}) should be positive. Symbolically, \[\frac{{\partial Q}}{{\partial L}} > 0,\frac{{\partial Q}}{{\partial L}} > 0\] - Inputs should be nonzero. Here, it means output is produced using positive amount of Labour and Capital. If any one of the input is zero then output will be zero. Symbolically, \[f(0,0) = f(L,0) = f(0,K) = 0\]
- Marginal Product of Labour and Capital are diminishing. Marginal Product remains positive but at diminishing rate, that is, each additional unit of input results in lower addition to output. Symbolically, it can be expressed as follows for Labour and capital respectively: \[\frac{{\mathop \partial \nolimits^2 Q}}{{\partial \mathop L\nolimits^2 }} < 0\] \[\frac{{\mathop \partial \nolimits^2 Q}}{{\partial \mathop K\nolimits^2 }} < 0\]
- As the amount Labour or Capital tends to be infinity, their Marginal Products tends to be zero. Symbolically, we can write:

\(\frac{{\partial f}}{{\partial L}} \to 0\) as \(L \to \infty \) and \(\frac{{\partial f}}{{\partial K}} \to 0\) as \(K \to \infty \)

It implies isoquants never touch axes and are strictly convex to the origin. It also implies diminishing returns to factor. - The production function should not specify a priori the degree of return to scale, however, the degree of return should be determined by the data.

The Neo-classical Production function is also called Well-behaved Production Function. Our discussion over the production function does not end here. We will discuss five classifications of the production function in a separate article. Your feedback is invited regarding the upcoming topic.

**Also Read:**

Theory of Production

4 Factors of Production: Methods, Resources and Inputs

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