# Law of Variable Proportions

The law states that if all other factors are fixed and one input is varied in the short run, the total output will increase at an increasing rate at first instance, be constant at a point and then eventually decrease.Marginal product will become negative at last.

According to G.Stigler, “As equal
increments of one input are added, the inputs of other productive services being held constant, beyond a certain point,the resulting increments of product will decrease, i.e., the marginal product will diminish”.

Assumptions:
The Law of Variable Proportions is based
on the following assumptions.

*Only one factor is variable while others are held constant.
*All units of the variable factor are homogeneous.
*The product is measured in physical units.
*There is no change in the state of technology.
*There is no change in the price of the product.

Total Product (TP)
It refers to the total amount of commodity produced by the combination of all inputs in a given period of time.
*Summation of marginal products, i.e.
TP = ∑MP
where, TP= Total Product, MP= Marginal
Product

Average Product (AP)
It is the result of the total product divided by the total units of the input employed.In other words, it refers to the output per unit of the input.
Mathematically, AP = TP/N
Where,
AP= Average Product
TP= Total Product
N= Total units of inputs employed

Marginal Product (MP)
It is the addition or the increment made to the total product when one more unit of the variable input is employed. In other words, it is the ratio of the change in the total product to the change in the units of the input. It is expressed as
MP=ΔTP/ΔN

where,
MP = Marginal Product
ΔTP = Change in total product
ΔN = Change in units of input
It is also expressed as
MP = TP (n) – TP (n-1)

Where,
MP = Marginal Product
TP(n) = Total product of employing nth unit of a factor
TP(n-1) = Total product of employing the
previous unit of a factor, that is, (n-1)th unit of a factor.

The Law of Variable Proportions is
explained with the help of the following schedule and diagram:
In table 3.1, units of variable factor (labour)
are employed along with other fixed factors
of production. The table illustrates that there  are three stages of production. Though total
product increases steadily at first instant,
constant at the maximum point and then
diminishes, it is always positive for ever.While total product increases, marginal product increases up to a point and then decreases. Total product increases up to the point where the marginal product is zero. When total product tends to diminish marginal product becomes negative. In diagram 3.1, the number of workers is measured on X axis while
TPL, APL and MPL are denoted on
Y axis. The diagram explains the three
stages of production as given in the above
table.

Stage I
In the first stage MPL increases up to
third labourer and it is higher than THE average product, so that total PRODUCT is increasing at an increasing rate. The tendency of total product to increase at an increasing rate stops at the point A and it begins to increase at a decreasing rate. This point is known as ‘Point of Inflexion’.

Stage II
In the second stage, MPL decreases up
to sixth unit of labour where MPL curve
intersects the X-axis. At fourth unit of
labor MPL = APL. After this, MPL curve
is lower than the APL. TPL increases at a
decreasing rate.

Stage III
Third stage of production shows that the sixth unit of labour is marked by negative
MPL, the APL continues to fall but remains
positive. After the sixth unit, TPL declines
with the employment of more units of
variable factor, labour.