### The Production Function, Explanation of the Law, Law of Returns to Scale

Introduction
The traditional theory resources used for the production of a product are known as factors of production. Factors of production are now termed as inputs which may means the use of the services of land, labour, capital and organisation.

The Production Function
The Production function expresses a functional relationship amidst quantities of raw materials and goods. It shows how and to what extent output changes with variations in raw materials during a specified period. In the words of Stigler “The production function is the name given to the relationship between rates of input of productive services and the rate of output of product. It is economist’s summary of technical knowledge.” It is expressed as follows.
__
Q =F (L,M,N,C,T), where Q stands for the output of a good per
Unit of time, L for labour, M__ for management of organisation, N for land or natural resources, C for Capital and T for given technology and F refers to the functional relationship.

Law of Variable of Proportions
If one input is variable and all other raw materials are fixed the concern’s production function exhibits the law of variable proportions. If the number of units of a variable factor is increased, keeping other factors constant, how output changes is the concern of this law.  As per Leftwich “The law of variable proportions states that if a variable quantity of one resource is applied to a fixed amount of other input, output per unit of variable input will increase but beyond some point the resulting increases will be less and less with total output reaching a maximum before it finally begins to decline.”
Postulations
This law is based on the below postulations.
1. It is feasible to alter the proportions in which the a range of factors are collective
2. Only one factor is erratic while others are held invariable
3. All units of the changeable factor are standardized
4. There is no variation in expertise
5. It presumes a short run condition
6. The produce is calculated in physical units, in quintals, tonnes etc.
7. The price of the produce is specified invariable
Explanation of the Law
To explain this law more clearly, let us construct a sketch. The TP curve first rises at an enhancing rate upto point A where its slope is highest. From point A upwards, the total product increases at a diminishing rate till it reaches its highest point C and then is starts falling. Point A where the tangent touches the TP curve is called the inflection point upto which the total product increases at an increasing rate and from where it starts increasing at a diminishing rate.
The average product curve AP and the marginal product curve MP also raise with TP. The MP curve reaches its maximum point D when the slope of the TP curve is the maximum at point A.
The maximum point on the AP curve is E where it coincides with the MP curve. This point also coincides with point B on the TP curve from where the total product starts a gradual rise.
When the TP curve reaches its maximum point C, the MP curve becomes zero at point F. When the TP starts declining the MP curve becomes negative, i.e. is below X axis.
The rising, the falling and the negative phases of the total, marginal and average products are in fact the different stages of the law of variable proportions which are discussed below.
Phase I
Increasing Returns – In this stage, the average product reaches the maximum and equals the marginal product when 4 workers are employed which is represented in the above diagram. This stage is portrayed here from the origin to point E where the MP and AP curves meet. In this stage the TP curve also increases rapidly.
The main reason for increasing returns in the first stage is that in the beginning the fixed factor is large in quantity than the variable factor. When more units of the variable factor are applied to a fixed factor, the fixed factor is used more intensively and production increases rapidly.
It is also explained in another way. In the beginning the fixed factor cannot be put to the maximum use due to the non-adequacy of sufficient units of the variable factor. But when units of the variable factor are applied in sufficient quantities, division of labour and specialisation lead to per unit increase in production and the law of increasing returns operate.
Another reason for increasing returns is that the fixed factor is indivisible which means that it must be used in a fixed factor, production increases more than proportionately. This cause points towards the law of increasing returns.
Phase II
Law of diminishing returns – In this stage, between phases I and III is the most significant stages of production. Phase II starts when the average product is at its maximum to the zero point of the marginal product. At the latter point, the total product is the highest. This is represented in the diagram as the phases EB and FC. Here land is scarce and is used intensively.
More and more workers are employed in order to have larger output. Thus the total product increases at a diminishing rate and the average and marginal products decline. Throughout this stage, the marginal product is below the average product. This is the only phase in which production is feasible and profitable. Hence it is incorrect to say that the law of variable proportions is another name for the law of diminishing returns. In fact, the law of diminishing returns is only one phase of the law of variable proportions.
Phase III
Negative Marginal Returns – Production cannot take place in the phase III either. For in this phase total product starts declining and the marginal product becomes negative. The employment of the last worker actually causes a decrease in total output.
The dotted line represented in the diagram, FC where the MP curve is below the x axis. Here the workers are too many in relation to the available land making it absolutely impossible to cultivate it. To the right of Point F, the variable input is used excessively. Therefore production will not take place in this stage.
Conclusion
Therefore the best stage of production is phase II, law of diminishing returns where the production and profitability both are feasible. Not only that, the production is optimum in this phase. Unlike other two phases, where the production is impossible and hence there is no profit generated.
Thus phases I and III are of economic absurdity or mere economic non-sense. Hence no manufacturer will produce in this phase by employing more units of the variable factor beyond the point zero marginal products MP because there is reduction in total product TP.

The Law of  Diminishing Returns
Benham defines the law of diminishing returns as “As the proportion of one factor in a combination of factors in increased, after a point the average and marginal product of that factor will diminish.”
It’s Application – Marshall applied the operation of this law to agriculture fisheries, mining, forests and the building industry. He has defined as “An increase in the capital and labour applied in the cultivation of land causes in general a less than proportionate increase in the amount of produce raised, unless it happens to coincide with an improvement in the arts of agriculture.”
Importance
1. In the words of Wick steed, the law of Diminishing returns “is an universal as the law lie itself.” The universal applicability of this law has taken economics to the realm of science.
2. It forms the basis of a number of a number of doctrines in economics. The Malthusian theory of population shoot from the fact that food supply does not amplify rapidly than the growth in population since the operation of the law of diminishing returns in agriculture. In fact this law was responsible for Malthus’ pessimism.”
3. Ricardo also based his theory of rent on this principle. Rent arises in the Ricardian sense because the operation of diminishing returns on land forces the application of additional doses of labour and capital on a piece of land does not increase output in the same proportion due to the operation of this law.
4. Like wise, the law of diminishing marginal utility in the theory of demand and that of diminishing marginal physical productivity in the theory of distribution are also based on the doctrine.
5. In under developed countries it is of an elementary importance for understanding their problems. In such economies agriculture is the main occupation of the people. The heaviness of populace on land enlarges with the augment in population. Consequently more and more persons are employed on land which is a fixed factor. This leads to declining marginal productivity of workers.
The Law of Returns to Scale
The law of returns to scale explains the relationship amidst outputs and the scale of inputs in the long run when all the inputs are increased in the same proportion. As per Roger Miller the law of returns to scale refers “to the relationship between changes in output and proportionate changes in all factors of production.” To meet a long run change in demand, the firm enhances its scale of production by using more space, more machines and labourers in the factory.