TYPES OF PRODUCTION FUNCTION

In economic theory we are concerned with three types of production functions, viz.;
? The production function with one variable input.
? The production function with two variable inputs and
? The production function with all variable inputs.
THE LAW OF VARIABLE PROPORTION
This law states that if equal increments of an input are added, and the quantities of other inputs held constant, the resulting increments of product will decrease beyond some point; that is, the marginal product of the input will diminish.
The law of diminishing returns is sometimes called the law of diminishing marginal returns to emphasize the fact that it deals with the diminishing marginal product of a variable input factor. The law of diminishing returns can’t be derived deductively. It is a generalization of an empirical regularity associated with every known production system.
The law of variable proportion (law of diminishing returns) is related to short run production process. It refers to the behavior of output as the quantity of one factor is increased; keeping the quantity of other factors fixed. Further it states that the marginal and average product will eventually decline.
Assumptions
It is assumed that there is no change in the technique of production
The law specially operates in the short-run
It is assumed that some inputs are kept fixed and labour is varied
This law is based on the assumption that the variable resources are applied unit by unit
The variable factors are homogeneous
Inputs prices remain unchanged, and
The law will not be applied if the proportions between factors are fixed for example: tailor master and sewing machine, computer operators and computers.

Therefore, the above assumptions must be satisfied for the application of the law of variable proportion. The situation of total product (TP), average product (AP) and marginal product (MP) can be explained with the law of variable proportion when quantity of variable factor is to the fixed factor of production.
The law can be illustrated with the help of an iron mine industry. Lets suppose the iron mine industry has a set of mining machinery as its capital (K) which is fixed in short-run and the industry can employ more of labour to increase its iron production.
The three stages
With labour time continuously divisible, we can smooth TP, MP and AP curves. The MPL (given by the slope of kthe tangent to the TP curve) rises upto point E’ becomes zero at J’, and is negative thereafter. The APL (given by the slope of the ray from the origin to a point on the TP curve) rises upto point F’ and declines thereafter (but remains positive as long as TP is positive). Stage I of production for labour corresponds to the rising portion of the APL. Stage II covers the range form maximum APL to where MPL is zero. Stage III occurs where MPL is negative.
Stage I: Stage of increasing returns
In this stage the following states are observed
? MP>AP i.e. marginal product is greater than average product
? After point E’, MR is decreasing but it is more than AP
? Upto point E, TP increase at an increasing rate, i.e. the slope of TP curve will be increasing. So, at E’, MP will be maximally and after that MT will start decreasing.
? Stage I ends after MP equals AP.
Causes of increasing returns in stage I
The causes of increasing returns in stage I is as follows
? In the initial stage the quantity of fixed facto is abundant in comparison to the quantity of variable factor
? Increases in the efficiency or productivity of variable factor.
Stage II: Stage of diminishing returns
In the stage the following states are observed
? AP and MP both will be decreasing
? AP> MP i.e. average product is greater than marginal product
? The stage ends when marginal product becomes zero i.e., MP = 0
? When the slope of the TP is zero, the MP is also zero. Since N’ and MP both are decreasing, this stage is known as the stage of decreasing returns.
Causes of diminishing returns in stage II
The causes of diminishing returns in stage II are as follows:
? Scarcity of the fixed factor relative to the quantity of the variable factor
? Indivisibility of fixed factor
? Imperfect substitutability of the factors
Stage III: Stage of negative returns
In this stage the following states are observed
? TP is declining.
? Slope of TP is negative, i.e. slope of TP (MP) to
? AP is declining but positive
Causes of negative returns in stage III
? If we keep on adding variable factors like labour on a given quantity of fixed factor (land), this will lead to overcrowding on the fixed factor. There will be therefore, lower availability of tools and equipment per worker, which causes fall in production.
? Use of too much of variable factors like labour also creates the problem of effective management. When there are too many workers, they may shift responsibility on to others. It becomes difficult to fix responsibility. The labour can therefore avoid the work.