# LAW OF VARIABLE PROPORTION

**Introduction**

Before we discuss the law of variable proportion or law of diminishing return, we wish to highlight (briefly explain) the following important terms:

**1) Fixed factor**

It is a factor that does not vary or change (neither increase nor decrease) with output. In the short-run, the fixed factors always remain constant irrespective of level of (changes in) output.

In Economics, short run means the time or period when something, especially a factory, an institution (a firm), etc, is newly established.

Good examples of fixed factors are:

- An area of land upon which a firm is built.
- Fixed capital goods: a plant, machines, vehicles, equipment, tools, etc.
- Management – director, manager, secretary, night guard, etc. It is also important to note that the costs incurred on these fixed factors are referred to as ‘Fixed Costs’.

**2) Variable Factor (VF)**

It is a factor that varies (changes) with output. As variable factor increases, output also increases, and. vice versa. Good examples of variable factors are: factory workers, number of casual labourer, raw materials, like flour and sugar in baking industry.

It is important to note that in the long run, all factors of production are termed as “variable factors” as all the factors can be changed – appreciably increased. For example:

- The parcel of land can be increased or the firm may be moved to a larger (permanent) site.
- The plants – machines, vehicles can be enlarged: new ones can be acquired and installed.

**3) Total Product (TP)**

TP is the “total quantity” of a commodity produced in a factory in a given period. TP depends entirely on the amount of variable factors employed (used). TP rises if the amount of variable factors increases, and vice versa.

**4) Average Product (AP)**

Briefly, AP is output per factor input; that is, output per one person. In other words, AP is the total product divided by the units of variable factor (labour -a number of persons) employed: The second definition indicates the formula for calculating AP. Symbolically, it is expressed as follows:

AP = Average Product.

TP = Total Product.

L = Labour.

VF = Variable Factor.

**5) Marginal Product (MP)**

- MP is addition to total product (total output) as a result of employing or using an additional unit of the variable factor, e.g. labour, raw material (a bag of flour), etc.
- MP is a change in output as a result of a change in the amount of the variable factor.
- MP is a change in TP due to a change, either a little more or a little less, of the variable factor.

Its formula

MP = Marginal product.

TP = Total product.

L = Labour.

VF = Variable factor.

**(Δ) **= means “change in”.

2. MP = Present TP (TP2) minus Previous TP (TP1).

MP = TP2 – TP1

For example, the employment of one more labour or uses of an extra bag of flour increases total product from 100 loaves to 130 loaves. Thus the MP is (130 – 100), 30 loaves of bread.

**Law of Diminishing Return**

The law of variable proportion is now briefly referred to as “Law of diminishing return”. The law states that:

**“If successive units of a variable factor of production e.g. labour, are applied to (combined with) a fixed factor of production, e.g. an acre of land, output increases at the first time, but after a certain time the addition of the unit of the variable factor (one more person) adds less to the total output than the preceding (previous) one”. **

At the beginning of the process of combining variable factor (people – farmers) with a fixed factor (a specific area of land like one acre), output increases. And this continues as more and more units of the variable factor (more people) are added to the fixed factor (one acre of land). But after sometime, the average, marginal and total output will start to decline (decrease).

It is important to note that the LVP (or law of return) embodies (contains) two laws; i.e. it is in two parts:

**Law of increasing return:** it operates when output is rising specifically when marginal product (MP) is rising,

**Law of diminishing (decreasing) return:** it operates when output is falling, specifically when MP starts to fall.

When output rises it is called “increasing return”. But when output decreases with increasing variable factor (s), it is called “diminishing or decreasing return”.

The following diagram shows the graphical illustration of both principles and laws. That is, it shows the two parts of law of variable proportion.

**Note**

The law of diminishing return is a part of the law of variable proportion. The law of diminishing return can best be explained with a means of production schedule (table) and a graph. Table 3 shows how output increases as the increasing quantities of a variable factor are added to a given quantity of the fixed factor.

**Example 1**

Determine the values of X and Y in table 3.

**Solution:**

Let us apply the above given formulas to calculate the values of X and Y in the table.

AP = TP ÷ L, X= 17÷9 = 1.88 = 1.9.

MP = TP2 – TP1; Y = 17 – 20 = -3.

You should note that the point where MP is at maximum indicates the ‘end of increasing return’.

And such a point (point E in figure 7) is called

- Point of diminishing marginal return, or
- Point of maximum marginal product.

The quantity of variable factor that corresponds’ with it is called:

- Optimum factor combination (OFC), or
- Optimum factor proportion.

OFC refers to the proportion of (different) variable factors which are combined with the fixed factor(s) for maximum output. In other words, OFC signifies (means) the maximum number of different quantities of variable factors to be combined with the fixed factor(s) that give optimum (maximum) output.

That is, OFC yields maximum output;, it yields the greatest number of goods that could be produced in a factory in a given period. Hence it always corresponds with the end of increasing return. It occurs at the point where MP is at the peak. And any increase in variable factor beyond such a point causes a decrease in output, Thus after this point the law of diminishing return normally sets in .

In the given example, table 3, OFC occurs at the 4th unit of labour. Thus the 5th person shouldn’t be employed as it makes MP to decline (fall). And note that it is not the point where TP or AP reaches maximum, like point D or F, that indicates OFC. OFC is solely determined or indicated by the peak of MP, point E in figure 7.

**Relationship between Marginal Product (MP), Average Product (AP) and Total Product (TP).**

- When MP curve is rising, AP curve lies below it MP curve lies above AP curve because: ‘the output of the additional worker is more than the average output of all the previous workers’.
- When MP curve is falling, AP curve lies above it. MP curve lies below AP curve because the output of the additional worker is less than the average output of all the previous workers.
- When AP is at its maximum, MP and AP are equal (MP = AP).
- When TP is at its maximum, MP is zero. It is because if output of an additional person is zero, TP will not rise. TP only increases if a new employee (an additional worker) adds more units to the previous total output.
- When TP curve is falling, MP has negative value. In other words, if MP has a negative value, TP falls. It is because the negative value of the MP causes reduction in TP.

**Assumption:**

A firm has 5 workers and its total output is 40 units, the average product is (40÷5) = 8 units.

It employs an additional worker (6th) person) who adds 10 units to total output, the TP is now 50 units and AP is (50÷6) = 8.3 units.

It employs another worker (7th person) who adds 7 units to total output, the TP is now 57 units. Thus the AP is (57 ÷ 7) = 8.1 units.

It employs additional worker (8th person) whose contribution is zero. That is, its employment does not add any thing (unit) to total output. TP remains the same; while AP declines to (57 ÷ 8) = 7.1 units.

It employs another person (9th person) whose contribution is negative (-2), and total output declines. This may occur because, probably, there is much congestion in the farmland or in the factory. And all the workers can’t work as effectively as before. This situation arises when diminishing return sets in.

The total product becomes (57 – 2) = 55 units, and average product is (55 ÷ 9) = 6.1 units.

We wish to summarize the above in a table, table 4, below.

**Implication (Outcome)**

If a new worker is relatively efficient (more efficient than the old workers), he raises the overall level of efficiency in a firm. Therefore, TP and AP rise. Conversely, if additional worker is relatively inefficient (less efficient than the old workers), he lowers the rate of productivity (reduces overall level of efficiency) in a firm. Thus TP and AP fail.

Interpretation of the various points in figure 7.

Point:

1)** ‘E’** is the point of diminishing marginal return (or point of maximum MP). It corresponds with point B on the TP curve. This indicates that when MP is at peak, TP is still rising. Point E also indicates optimum factor combination which is 4 units of labour.

2)** ‘F’** is .the point of diminishing average return (or point of maximum AP). It is also a point at which AP equals MP, or the point at which AP curve intersects MP curve. Secondly point F corresponds with point C on the TP curve. It indicates that when AP is at maximum, TP is still rising. Note that the quantity of a variable factor that corresponds with this point F is the quantity at which AP is at a maximum. Thus 5th (unit of labour (and not the 4th) is the unit of labour at which AP is at maximum.

3) **‘A’** corresponds with one (1) unit of the variable factor. At this point or at one unit of variable factor, TP, AP and MP are all equal at 2 output (see table 3 given above).

4) **‘G’** marks the point of intersection of the MP curve with the horizontal axis. Thus it is a point at which MP is at zero; and it corresponds with point D on the TP curve. And it therefore indicates that when TP is at maximum ,. MP is at zero. And the quantity of the variable factor (7th unit of labour and not the 6th unit) that corresponds with TP when MP is at zero indicates the point of maximum TP.

5)** “H to I”** indicates increasing return to scale. As the variable factor increases, TP, AP and MP — rise. This zone (area) witnesses increasing return.

6) **“I to J”** indicates diminishing return to scale. As the variable factor rises, output doesn’t rise proportionately. MP falls, AP rises and falls, while TP rises. This zone witnesses diminishing return.

7) **“J to K”** indicates negative return with regard to MP. This portion witnesses a continuous fall in TP. We recall that as TP falls, MP has negative value. That is, as soon as TP starts to fall, MP begins to have negative value. Thus point ‘L’ on the falling part of TP curve corresponds with point M located on the negative portion of MP curve.

**Example 2**

Determine the value of P to Z in the table given above.

Answer

Let us apply the formulas given above to calculate the values of P to Z as follows:

1) Value of P = TP÷L = 20 ÷ 3 = 6.7.

2) Value of Q = TP2 – TP1 = 20 – 11 = 9.

3) Value of U = 40 – 2 = 38.

Note that at 6″ unit, TP is 40, while the MP is 2. U is the previous TP.

TP2 – U = 2, 40- U = 2, 40 – 2 = U = 38.

4) Value of V:

Since we have got TP for the 5th unit, we can therefore calculate the value of V

V = AP : 5 = 38 : 5 = 7.6.

5) Value of R:

Note that U – R = 8, U = TP2 and R = TP1 that is, TP2 – TP1 = 8

therefore, 38 – R = 8, R = 38 – 8 = 30.

6) Value of S:

S = 30 ÷ 4 = 7.5

7) Value of T:

T = TP2 – TP1

T = 30 – 20 = 10

8) Value of W:

W = TP2 – TP1

W = 40 – 40 = 0.

9) Value of X:

38 ÷ 8 = 4.75 = 4.8

10) Value of Y :

Y= TP2 – TPI

Y = 38 – 40 = – 2.

11) Value of Z :

Z = 3.8 × 9 = 34.2 = 34

These answers to the values of P to Z have been filled in the appropriate spaces in the answer table given above.

**Importance of law of diminishing returns**

**Proper combination of factors of production:**The law of diminishing returns helps the entrepreneur to combine properly the factors of production to prevent wastage.**Changes in scale of production:**The law of diminishing returns helps entrepreneurs to change the scale of production through the variation of the quantities of all input.**It ensures efficiency:**As more and more variable factors are added to a fixed factor, it eventually comes to a profitable level and productivity and efficiency are maintained.