Returns to Scale

Introduction 

This topic is a part of study of production function. A production function is an expression of quantitative relation between change in inputs and the resulting change in output. It is expressed as;

Q = f (i1 , i2 ......in )

Where Q is output of a specified good and i1 , i2 ….in are the inputs usable in producing this good. To simplify let us assume that there are only two inputs, labour (L) and capital (K), required to produce a good. The production function then takes the form :

Q = f (K,L)

In microeconomics, conventionally, we study two aspects of relation between inputs and output. One aspect is : in what manner the change takes place in output of a good, if only one of the inputs required in producing that good is increased, i.e. other inputs kept unchanged? The manner of change in output is summed up in the law of variable proportions which you have already studied. The second aspect is : in what manner the output of a good changes, if all the inputs required in producing that good are increased simultaneously and in the same proportion. This aspect is technically termed as returns to scale, and is the subject matter of this study. The word 'return' refers to the change in physical output. The word 'scale' refers to the scale of operation expressed in terms of quantum of inputs employed.

Meaning

Returns to scale means the manner of change in physical output caused by the increase in all the inputs required simultaneously and in the same proportion. Elaborating, suppose one unit of capital and one unit of labour (1K + 1L), produce 100 units of output. Further suppose that both the inputs are doubled, i.e. 2K + 2L. The point of interest is : will output increase by just 100%; by more than 100%, or by less than 100%. There is no unique answer. All the three states are possible. The three states are respectively called Constant Returns to Scale (CRS), Increasing Returns to Scale (IRS) and Decreasing Returns to Scale (DRS). Let us first illustrate the three states and then explain reasons.

Constant Returns to Scale (CRS)

Suppose 1K+1L produce 100 units of output, and 2K+2L produce 200 units of output. It is 100 percent increase in inputs leading to just 100 percent increase in output. This manner of change in output is called CRS.

Increasing Returns to Scale (IRS)

Suppose 1K+1L produce 100 units of output and 2K+2L produce 250 units of output. It is 100 percent increase in inputs in leading to 125 percent increase in output. This manner of change in output is called IRS.

Decreasing Returns to Scale (DRS) 

Suppose 1K+1L produce 100 units of output, and 2K+2L produce 180 units of output. It is 100 percent increase in inputs leading to only 80% increase in output. This manner of change in output is called DRS. 

Which of the above states actually results depends to a great extent on the type of technology used. There are technologies which result in IRS from the beginning and continue upto a large output level. Similarly, there are technologies leading to CRS almost throughout. There can also be technologies leading to DRS from the very beginning.

Why do IRS arise?

There are two possible reasons:

1. More division of labour 

Division of labour means subdividing a task into many small sequential operations, with each worker (or a group of workers) assigned each operation. A single worker, instead of doing all the operations, concentrates on only one operation and specializes. This raises efficiency of the worker. 

Returns to scale means increasing the number of workers along with other inputs. More workers mean more division of labour. If one task can be divided into 20 small operations, with each worker assigned only one operation, the worker becomes an expert in the operation he is assigned. Efficiency increases and so the production. In business circles, the division of labour type production is called assembly line production.

2. Use of specialized machines

More capital means more capital goods and bigger capital goods. Fully automatic machines can replace the semi-automatic or the hand operated machines. Bigger machines can be used in place of small machines. Bigger capital goods can be used in place of smaller capital goods. It is a common knowledge that a double size capital input may produce more than double the output. Let us take an interesting example. Suppose a firm needs a wooden box to store goods. 

Suppose initially the firm goes in for 1'x1'x1' (LxBxH) size box. Let us see the input requirement and the resulting output. Let the wood be the only input required. A box has 6 sides. Each side requires 1 sq. ft. of wood (=1'x1'). Then

the input requirement = 1'x1'x6 = 6 sq.ft. 

The storing capacity of the box is measured by its volume. Then : 

Output of the box : 1'x1'x1' = 1 cubic ft. 

Let us now see what happens when the size of the box is increased to 2'x2'x2'. 

Input requirement = 2'x2'x6 = 24 sq.ft.

Output = 2'x2'x2' = 8 c.ft.

Now compare. Input of the box rises from 6 sq.ft. to 24 sq.ft. i.e. by 300%. Output of the box rises from 1c.ft. to 8 c.ft., i.e. by 700%. Increasing returns to scale arise. 

Remember that it may not go on for ever, i.e. we go on increasing the size and continue to get IRS. A stage may reach when IRS may give way to CRS or DRS.

Why do DRS arise?

Economists do not find any specific reason. DRS is a puzzle. Why output rises in a smaller proportion when all inputs are increased? The probable explanation is that the firm finds it difficult to manage and coordinate the activities arising out of larger scale. The difficulties may lead to wastage, inefficiency etc. and cause DRS.