Law of Variable Proportions:

"in a given state of technology, when the units of  variable factor of production (L) are increased within the units of other fixed factors, the marginal productivity increases at increasing rate up to a point, after this point. it will become less and less"

Assumptions:

The assumptions of the law of variable proportion are given as below:

1. It is assumed that the technique of production should remain constant during production.
2. It operates in the short-run because in the long run, fixed inputs become variable.
3. Some inputs must be kept constant.
4. The various factors are not to be used in rigidly fixed proportions but the law is based upon the possibility of varying proportions. It is also called the law of proportionality.
5. It is assumed that all the units of variable factors of production are homogeneous in amount and quality.
6. It is assumed that labor is a single variable factor.

Schedule:

The law of variable proportion is explained with the help of the following schedule:

In the above schedule, units of variable factor (labor) are employed with other fixed factors of production. The marginal productivity of labor goes on increasing up to the 3rd worker. This is so because the proportion of workers to other fixed factors was at first insufficient. After 3rd worker the marginal productivity goes on falling onwards till it drops down to zero at the 6th unit of labor. The 7th worker is only a cause of obstruction to the others and is responsible in making the marginal productivity negative. The marginal productivity (MP_L) and the average productivity (AP_L) equalize at 4the worker. Then the MP_L falls more sharply

Diagram:

The number of workers are measured on X-axis while TP_L, AP_L and MP_L on Y-axis. The above diagram shows the three stages also obtained from the schedule.

Stage I:

At this stage MP_L increases up to 3rd worker and its curve is higher than the average product, so that total product is increasing at increasing rate.

Stage II:

At this stage, MP_L decreases up to 6th unit of labor where MP_L curve intersects the X-axis. At 4the unit of labor MP_L = AP_L after this, MP_L curve is lower than the AP_L. TP_L increases at decreasing rate.

Stage III:

At 6the unit of labor the MP_L becomes negative, the AP_L continues falling but remains positive. After the 6th unit, TP_L declines with the employment of more units of variable factor (L).

Relationship Among Total, Average and Marginal Product:

The relationship among total, average and marginal product of labor in the light of the law of variable proportion is explained as under:

1. The marginal productivity of labor increases, the TP_L also increases at increasing rate. It is shown in the schedule up till 3rd unit of labor. The MP_L curve has positive slope and TP_L curve has rising tendency towards Y-axis.
2. When the MP_L decreases onwards till it drops to zero, the TP_L increases at decreasing rate as shown in the stage II and the TP_L curve has positive slope but has rising tendency towards X-axis
3. When the MP_L is equal to zero, the TP_L is maximum as shown on the 6th unit of labor.
4. When the MP_L becomes negative, the MP_L curves falls below the X-axis, the TP_L declines from its maximum position and its slope becomes negative as shown in the stage III in the above diagram.
5. When the MP_L increases, The AP_L also increases but at slow rate. The MP_L curve becomes above the AP_L curve. Both have positive slopes.
6. At some point, MP_L = AP_L. At this point, MP_L curve intersects the AP_L curve as shown at the 4th unit of labor in the above diagram.
7. After intersecting point, MP_L falls sharply. The MP_L curve becomes below the AP_L curve. Both curves have negative slope.
8. When MP_L becomes negative, the APL never becomes negative because it is calculated from the TP_L. So MP_L curve is below the X-axis but AP_L curve is above the X-axis, having negative slope.