BY : Arjun Paudyal

Production Function :

Production function may be defined as the functional relationship between physical inputs (i.e., factor of production) and physical outputs (i.e., the quantity of good produced). Production function shows technological or engineering relationship between output of a commodity and its inputs. The act of production involves the transformation of input into output. The word production in economics in not merely confined to effecting physical transformation in matter, it also covers the rendering of services. The production function can be expressed symbolically as,

X=f (Ld, L, K, M, T)

Where X denotes commodity X, Ld land, L labor, K capital, M management and T technology. The above function shows the general production function.

There are two types of production function:

1. Short run production function.

2. Long run production function.  

Long run production function :

Long run is the period where the fixed factor is changed. If the demand for the firm's product increases, the firm can increase its output by enlarging the size of its plant or increasing the scale of its operations. So in the long run there is enough time to effect changes in the scale operations or to introduce other adjustment in the organization set- up of the firm. In fact the firm in the long period, can build any desired scale of plant. All factors are variable none is fixed. In the long run, then, there are number of decisions that a firm will have to make about the scale of its operations, the location of its operations and the techniques of production it will use. In this concept it explains the laws of return to scale.

Laws of return to scale:

The laws of return to scale is long run concept. In the long run volume if production can be changed by changing all factor of production. It shows the behavior of output when all factor are altered in the same proportion. To understand laws of return we have to know about Isoquants.

Isoquants :

It is also known as Equal production curve. It has been developed to study the theory of production and to show the equilibrium of a producer regarding combination of factor. Iso-product curve, which is a parallel concept to the indifference, curves in the theory of consumption. Isoquant shows all possible combination of the two inputs physically capable of producing a given level of output. Any point on the isoquant is a recipe for the same output as any other point on the same curve.

Isoquant can be easily understand from table given below:

Equal product combination.

To begin with, combination A, representing 1 unit of factor X and 12 units of factor Y, produce a given quantity (say, 40 units) of a product. All others combination in the table are assumed to yield the same amount, i.e. 40 units of product.

{{ Figure Here }}

In the given figure IP depicts such iso-product curve on which are represented the various combination A, B, C, D, & E of the above table. IP represent all these combination with which 40 units of the product can be produced. The shape of the isoquants shows the degree of substitutability between the two factors used in production.

Properties of isoquants :

1. Isoquant slopes downwards from left to right. Because, when quantity of input labor is increased, the quantity of capital must be reduced so as to keep output constant.
2. Higher isoquant represents higher level of output. Because, higher isoquant consists of more of both labor and capital.
3. No two isoquants intersect each other. If two isoquants intersect each other it would mean that with the same unit of labor and capital, two different levels of output can be produced, which is absurd.
4. Isoquants are convex to the origin due to diminishing marginal rate of technical substitution.

Returns to scale are of three kinds- constant, increasing and decreasing return to scale.

Return to Scale

1. Constant Return to Scale:

If we increase all factors in a given proportion and the output increase in the same proportion, returns to scale are said to be constant. Thus, if a doubling or trebling of all factors causes a doubling or trebling of output, returns to scale are constant. The Constant Return to scale can be illustrate from the figure below:

{{ Figure Here }}

In figure it is assumed that, in the production of the good, only two factors, labor and capital are used. The straight ling passing through the origin indicate the increase in scale as we move upward. It is seen from the figure that successive isoquants are equidistant from each other along the straight line drawn from the origin. The distance between the successive equal product curves being the same along the straight line through the origin means that if both labor and capital are increased in a given proportion, output expands by the same proportion. Thus this figure display constant return to scale

Causes:

1. Divisibility :

If the factors of production are perfectly divisible production function must necessarily exhibit constant returns to scale. It implies that the constant returns to scale may not occur only if the factor cannot be increased or decreased in same proportion.

• Increasing Return to Scale :

Increasing return to scale means that output increase in greater proportion than the increase in inputs. For example: If increasing input by 50% lead to an increase in production by 70%, the return to scale is said to be increasing. This can be shown through isoquant map which is given below:

{{ Figure Here }}

In the given figure the various isoquants Q1, Q2, Q3 are drawn which successively represent 100, 200 and 300 units of output. It will be seen that distance between the successive isoquants decrease as we expand output by increasing the scale. Thus increasing return to scale occur since OA>AB>BC which means that equal increase in output are obtained by smaller and smaller increments in inputs.

Causes :

a. Indivisibility:

Some factors are available in large and lumpy units and can therefore be utilized with utmost efficiency at a large level of output. Thus, in the case of some indivisible and lumpy factors, when output is increased from

a small level to a large one, indivisible factors are better utilized and therefore increasing returns are obtained.

1. Specialization :

If the factors were perfectly divisible, with the increase in the scale, returns to scale can increase because the firm can introduce greater degree of specialization of labor and machinery and also because it can install technologically more efficient machinery.

• Decreasing Return to Scale:

When output increases in a smaller proportion than the increase in all inputs, decreasing returns to scale are said to prevail. Example: If increasing input is by 50% leads to increase in production by less than 50%. The return to scale is said to be decreasing. This case can be shown on an isoquant map given below:

In given figure successively decreasing return to scale occur since A>OA and BC>AB and CD>BC. It means that more and more of input are required to obtain equal increase in output.

{{ Figure Here }}

Causes :

1. Indivisible entrepreneur :

According to this view the entrepreneur is a fixed factor of

production, which all other inputs may be increased, he cannot be. So, the law of diminishing return operates. Thus they point out that we get diminishing returns to scale beyond a point because varying quantities of all other inputs are combined with a fixed entrepreneur.

1. Difficulty for management :

When the firm has expanded to a too gigantic size, it is difficult to manage it with the same efficiency as previously.

Varying Returns to Scale in a Single Production Process

It should be noted that it is not always the case that different production function should exhibit different type of return to scale. It generally happens that there are three phases of increasing, constant and diminishing return to scale in a single production function. In the beginning when the scale increases, increasing return to scale are obtained because of greater possibilities of specialization of labor and machinery. After a point, there is a phase of constant return to scale where output increase in the same proportion as input. If the firm continues to expand, then eventually a point will be reached beyond which decreasing return to scale will occur due to the mounting difficulties of co-ordination and control.

{{ Figure Here }}

In given figure upto point C in a ray OR from the origin, the distance between the successive isoquants showing equal increments in output goes on decreasing. This implies that upto point C equal increments in output are obtained from the use of successively smaller increases in inputs. Thus, upto point C on ray OR increasing returns to scale occur. Further, it will be seen that from point C to point E constant returns to scale are obtained as the same proportionate increments in output are obtained from the proportional increase in inputs. Beyond point E, the distance between the successive isoquants representing equal increments in output is decreasing along the ray OR from the origin which implies that less than proportionate increase in output is obtained from the same proportionate increments in the use of the factors.

Assumption :

• Two axis represent two variable inputs:

It is assumed that two axis of the curve represent two variable inputs because this law is based on the behavior of outputs in relation to change in scale.

• State of technology is fixed:

It is assumed that the state of technology remains unaltered.

• Proportion of inputs is constant but the scale of return is different:

It is assumed that the proportion of inputs is constant but the scale of return is different because increase in scale thus occurs when all factors or inputs are increased keeping factor proportions unaltered.

• All inputs are homogenous:

It is assumed that all inputs are homogenous.

Conclusion:

We can say that laws of returns to scale are a matter of interaction between economies and diseconomies of large scale production. Initially when a firm expands, it faces increasing returns to scale because of the scale economies. As the scale of operation rises, increasing return to scale give way to constant returns to scale, because here economies and diseconomies of large scale production balance each other. But if the firm continues to expand its scale of production beyond a point, it experiences diminishing returns to scale. This is due to the fact that eventually the economies of large scale production are swamped by the diseconomies of large scale production and this results in decreasing returns to scale.

References:

1. Dr. Shyam Joshi
2. H.L. Ahuja
3. K. W. Dwett