## Law of Variable Proportions/ Law of Diminishing Returns

 The law of variable proportion is one of the fundamental laws in economics. This law deals with the short-run production function. In short run, input and output relations are studied by keeping at least some factors/ inputs of production constant. The law of variable proportion states that when more and more unit of a variable input are applied with the given
number/size of the fixed inputs, the total output may initially increase at an increasing rate up to some extent, then it increases at decreasing rate, it reaches to the maximum level of ultimately it starts to decrease. This tendency of production in the short run is termed as the ‘Law of Variable Proportion’ or the Law of Diminishing Return’. This law of production exhibits the direction of the change of total product and the marginal product of the variable factor combined with fixed factors.

Different economists have offered definitions of the Law of Variable Proportion or the Law of Diminishing Return in their own way.

Alfred Marshall discussed the law of diminishing returns in relation to agriculture. He defines the law in these words “An increase in the capital and labor 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.”
Similarly, according to Prof. Leftwitch, “The law states that if the input of one resource is increased by equal increments per unit of time while the inputs of other resources are held constant, TP will increase, but beyond some point the resulting TP will increases will become smaller and smaller.”
From the definitions presented above, it is clear that the law of variable proportions or the law of diminishing returns refers to the behavior of output as the quantity of one factor is increased keeping the quantity of other factors fixed, and further it states that the marginal and average product will eventually decline.

The law of variable proportions has the following assumptions:
1. Constant Technology: The law of variable proportion assumes that the state of technology is constant. The reason is that if the state of technology changes the marginal and average productivity of variable may rise instead of diminishing because of technological improvement.
2. Short-run: This law specially operates in the short-run because in the short run some factors are fixed and the proportion of others has to be varied. It assumes that labor (L) is variable factor while capital (K) is the fixed factor.
3. Homogeneous Factors: This law is based on the assumption that the variable factor (labor) is applied unit by unit and each factor unit is homogeneous/identical in amount and quality.
4. Changeable Input Ratio: The law supposes that it is possible to produce output by changing the ratio of factor inputs; the ratio of fixed and variable factor is changeable; there is no fixed proportion production function.
In the short period of time capital is held constant in manufacturing while in agriculture land is held constant and other inputs are used in varying number. The law of variable proportions or the law of diminishing returns is illustrated with the help of hypothetical data given in table. In the illustration labor service is supposed as a variable factor.
 No. of Labor Total Product Marginal Product Average Product Stage of Returns (L) (TPL) APL=TPL/L MPL=∆TPL/∆L 1 2 3 4 5 6 7 8 9 10 11 8 22 39 52 60 66 70 72 72 70 66 8 11 13 13 12 11 10 9 8 7 6 8 12 17 13 8 6 4 2 0 -2 -4 1st Stage 2nd Stage 3rd Stage
The data in table, show that production changes due to change in variable factor (labor). As the number of labor is increased, initially, both marginal and average product of labor increases. After employing more of labor, both APL and MPL falls more rapidly. Falling in the APL and MPL will continue as more labor is put in the production. Hiring of 9th unit of labor adds only nominal amount of output on iron industry. After then, the additional unit of labor i.e., 10th unit, the marginal product becomes negative. Here after, production becomes less.

The operation of the law of variable proportion can be explained in the following figure.
 Three Stages of Variable Proportions
The vertical axis of the both figure shows the total, average and marginal product of the variable factor and the horizontal axis shows the units employed of the variable factor. The quantity of variable factor is increased relative to the fixed factors, there may arise three different stages.

### First Stage

This stage covers the production ranges OL2 units of labor. In the first stage, total product (TP) increases at increasing rate up to the point A (i.e., called point of inflexion) after then TP increases at diminishing rate. At that movement of operation MPL increases till the use of OL1 unit of labor and begins to decline whereas APL continuously increases till OL2 unit of labor. In this stage, MPL is greater than APL. MPL and APL become equal at OL2 units of labor employed, as shown by the point B’. Point B is the end of this stage and the second stage starts.

Causes of returns in stage first

1. Increase in efficiency of fixed factor: In the initial stage, the quantity of fixed factor is abundant in comparison to the quantity of variable factor. As more units of variable factors are added to the constant quantity of fixed factor than the fixed is more intensively and effectively utilized i.e., the efficiency of fixed factors are added to it.
2. Increase in efficiency: In the initial stage, we get increasing returns because as more units of the variable factors are employed, the efficiency of the variable factors itself increases. The reason of efficiency is that with sufficient quantity of variable factor introduction of division of labor and specialization becomes possible which result in higher productivity.

### Second Stage

This stage covers the production ranges between OL2 and OL3 units of labor. In this stage, both MPL and APL decline but positive. However, MPL declines at the faster rate. It is important to note that, at OL3 unit of labor TP becomes maximum, as shown by the point M in penal (A) and MPL becomes zero level at point OL3.

Causes of returns in stage second

1. Scarcity of fixed factor: In the short-run, the quantity of fixed factor cannot be varied. For this reason, the further increases in the variable factor will cause marginal and average product to decline because the fixed factor then becomes inadequate relative to the quantity of variable factor.
2. Indivisibility of fixed factor: If the fixed factors are perfectly divisible, there no change in proportion or increasing and decreasingly returns. In short-run, the size of plant is being unchanged, so the fixed factors are indivisible. Therefore, the excess variable factor are used in combination with indivisible fixed factor, the average product of variable factor diminishes.
3. Imperfect substitutability of the factors: The operation of law of diminishing returns is the imperfect substitutability of one factor for another. The perfect substitute of the scarce fixed factor been available, then the paucity of the scarce fixed factor during the 2nd stage would have been made up by increasing the supply of the perfect substitute with the result that output could be expended without diminishing returns.

### Third Stage

Third stage begins with the decline in TPL. As the figure shows, the use of labor in excess of OL3 cause decline in TPL and negative MPL because of overcrowding of labor. However, APL will be positive i.e., tends to zero but not becomes zero.

Causes of returns in third stage

1. Too excessive amount of fixed factor: As the amount of the variable factor continue to be increased to constant quantity of the other; a stage is reached when the total product declines and marginal product become negative. This is due to the fact the quantity of variable factor becomes too excessive relative to the fixed factor so that they affect each other’s efficiency.