Long run production function – Law of returns to scale
Long run refers to the period of time over which it is possible to vary the inputs of all factors of production. Thus in the long run all the factors of production becomes variable. Hence there is no distinction between fixed factors and variable factors. The relationship between inputs and outputs, in the long run, is known as “Law of Returns to Scale”.
Statement
“Other things being equal, as a firm, in the longrun, increases the quantities of all factors employed, the output may rise initially at a more rapid rate then the rate of increase in input, then the output may increase in the same proportion of input and ultimately the output increases less proportionately”.
There are 3 phases of returns to scale. They are:
 Increasing returns to scale
 Constant returns to scale
 Decreasing returns to scale.
As we all know that production function explains us the relationship between input and output. When we say that we are using different factors of production (inputs), we should note here is that, we are going to use different factors of production in varying proportions. The Law of Returns to scale arises, only when we use the combination of various factors of production in varying proportions.
Assumptions
 For the convenience of study we take only three factors of production. Viz., land, labour, capital.
 In this, land will be treated as fixed or constant factor.
 Labour and capital are the varying factors i.e., we are going to vary the labour and capital proportion and study the output behaviour.
 We are going to increase the labour proportion by 2 units and increase capital by 1 unit.
This concept could be better explained through a numerical schedule:
Total Output 
Total Output 
Marginal Output 
1 (1+2) 
50 
50 
2 (2+4) 
110 
60 
3 (3+6) 
180 
70 
4 (4+8) 
250 
70 
5 (5+10) 
300 
50 
6 (6+12) 
335 
35 
7 (7+14) 
350 
15 
Observations
 As we go on increasing the proportion of K & L in the production, the total output is increasing but at a diminishing rate.
 The Marginal output which is most important here
 Increases at a rapid rate then the rate of increase in the input – in the 1st stage. Hence it is increasing returns to scale.
 Increases in the same ratio as increase in the input – in the 2nd stage. Hence it is constant returns to scale.
 Decreases as you employ more and more of K & L – in the 3rd stage. Hence it is decreasing returns to scale.
These three concepts may be explained through a diagram also.
Stage 1: Increasing returns to scale
When the increase in output is more than proportional to the increase in input, it is referred to as the law of increasing returns to scale. Increasing returns to scale implies decreasing average costs. In the above table, we have assumed that labour and capital are the only two inputs.
If say, labour and capital are increased by 15% and output increases by 30%, we say that there is increasing returns to scale.
The application of increasing returns to scale is due to the reason that as the firm expands production, it gets many advantages, known as economies of scale, like better division of labour, technical and managerial economies, etc. So, it can produce additional output with lesser inputs than before, thus enjoying increasing returns to scale.
Stage 2: Constant returns to scale
When the increase in output is proportional to the increase in inputs, we say that there are constant returns to scale. Here, an increase in the capacity of the firm has no effect on the long run average cost of production. A constant return implies constant cost.
If say, there is 5% increase in all the factors, it will result in an equal proportion of 5% increase in the output. Here, the Marginal Product is constant. Constant returns to scale are also called “linear homogeneous production function”.
The application of constant returns to scale is due to the reason that as a firm increases its output, a stage comes when all the economies have been fully exploited. Now, the expansion of the output is in the same proportion as that of the input, giving rise to constant returns to scale.
Stage 3: Decreasing returns to scale
When the increase in the output is less than proportional to the increase in inputs, we say that there is decreasing returns to scale. Decreasing returns implies increasing average costs.
For example, if all the factors are increased by 5%, the output will increase by less than 5%, say, by 3%. In this phase, marginal product will decrease.
The stage of diminishing returns applies when, with further expansion, diseconomies of scale, like lack of coordination, difficulties of management arise. Hence, due to diseconomies of scale, the management has to use inputs in greater proportions, thus giving rise to decreasing returns to scale.
Note:
Stage1 Increasing returns to scale occur when the % change in output > % change in inputs
Stage2 Constant returns to scale occur when the % change in output = % change in inputs
Stage3 Decreasing returns to scale occur when the % change in output < % change in inputs
A numerical example of long run returns to scale
Units of Capital 
Units of Labour 
Total Output 
% Change in Inputs 
% Change in Output 
Returns to Scale 
20 
150 
3000 



40 
300 
7500 
100 
150 
Increasing 
60 
450 
12000 
50 
60 
Increasing 
80 
600 
16000 
33 
33 
Constant 
100 
750 
18000 
25 
13 
Decreasing 