Solution:

The Law of Variable Proportions:

• If one input is variable and all other inputs are fixed the firm's production function exhibits the law of variable proportions. If the number of units of a variable factor is increased, keeping other factors constant, how output changes are the concern of this law.

• Suppose land, plant, and equipments are the fixed factors and labor the variable factor. When the number of laborers increases successively to have larger output, the proportion between fixed and variable factors is altered and the law of variable proportions sets in.

• The law states that as the quantity of a variable input is increased by equal doses keeping the quantities of other inputs constant, the total product will increase, but after a point at a diminishing rate.

• This principle can also be defined thus: When more and more units of the variable factor are used, holding the quantities of fixed factors constant, a point is reached beyond which the marginal product, then the average, and finally the total product will diminish.

• The law of variable proportions (or the law of nonproportional returns) is also known as the law of diminishing returns. But as we shall see below, the law of diminishing returns is only one phase of the more comprehensive law of variable proportions.

Assumptions:

• The law of diminishing returns is based on the following assumptions:

  1) Only one factor is variable while others are held constant.

  2)All units of the variable factor are homogeneous.

  3)There is no change in technology.

  4)It is possible to vary the proportions in which different inputs are combined.

  5)It assumes a short-run situation, for, in the long run, all factors are variable.

• The product is measured in physical units, i.e. in quintals, tones, etc. The use of money in measuring the product may show increasing rather than decreasing returns if the prices of the product rise, even though the output might have declined.

Explanation:

• Given these assumptions, let us illustrate the law with the help of Table 1, where on the fixed input land of 5 acres, units of the variable input labor are employed and the resultant output is obtained.

• The production function is revealed in the first two columns. The average and marginal product columns are derived from the total product columns.

• The average product per worker is obtained by dividing column (2) by a corresponding unit in column (1).

• The marginal product is the addition to the total product by employing an extra worker. 3 workers produce 36 units and 4 produce 48 units.

  

• Products increase at first, reach a maximum, and then start declining. The total product reaches its maximum when 7 units of labor are used and then it declines.

• The average product continues to rise till the 4 the unit while the marginal product reaches its maximum at the 3rd unit of labor, then they also fall.

• It should be noted that the point of falling output is not the same for total, average and marginal products. The marginal product starts declining first, the average product following it and the total product is the last to fall.

• This observation points out that the tendency to diminishing returns is ultimately found in the three productivity concepts.

• The law of variable proportions is presented diagrammatically in Figure. The TP curve first rises at an increasing rate up to point A where its slope is the highest.

• From point A upwards, the total product increases at a diminishing rate till it reaches its highest point C, and then it starts falling.

• Point A where the tangent touches the TP curve is called the inflection point up to which the total product increases at an increasing rate and from where it starts increasing at a diminishing rate.

• (MP) and the average product curve (AP) also rises with TP. the MP curve reaches its maximum point D when the slope of the TP curve is the maximum at point A.

• The maximum point on the AP curves is E where it coincides with the MP curve. This point also coincides with point B on the TP curve from where the total product starts a gradual rise.

• When the TP curve reaches its maximum point C the MP curve becomes zero at point F. When TP starts declining, The MP curve becomes negative. It is only when the total product is zero that the average product also becomes zero.

• The rising, the falling and the negative phases of the total, marginal and average products are the different stages of the law of variable proportions which are discussed below.

Returns to Scale:

• Returns to scale describe the relationship between outputs and scale of inputs in the long run when all the inputs are increased in the same proportion.

• In the words of Prof. Roger Miller, "Returns to scale refer to the relationship between changes in output and proportionate changes in all factors of productions." To meet a long-run change in demand, the firm increases its scale of production by using more space, more machines, and laborers in the factory.

Assumptions:

• This law assumes that,

  1)All factors (inputs) are variable but the enterprise is fixed.

  2)A worker works with given tools and implements.

  3)Technological changes are absent.

  4)There is perfect competition.

  5)The product is measured in quantities.

Explanation:

• Given these assumptions, when all inputs are increased in unchanged proportions and the scale of production is expanded, the effect on output is three stages: increasing returns to scale, constant returns to scale, and diminishing returns to scale. They are explained with the help of Table 2.