This is an article on what the law of variable proportion is all about. You will learn the core concept of this very famous production theory in economics. Before reading this article, we recommend you read through the fundamental concept regarding production and production function, which will facilitate your learning vigor ahead.

- What are the main assumptions underlying the law of variable proportion?
- What does the law of variable proportion state?
- Numerical and graphical explanation of the law of variable proportion
- Stage of operation for a rational producer
- Application of the law of variable proportion

As you have studied the recommended article, I guess so, you know very well the fundamental concept of production function. Let’s go ahead with the theory.

For the sake of simplicity, we assume that the production function consists of only two-factor inputs- labor and capital. Accordingly, the production function turns out to be Q = f (K, L), which includes only two factors – K and L – by dropping out other factors. This is because K and L are the dominant factor inputs and simplify the explanation as well as the graphical presentation of the theory.

Holding at least one-factor input constant, ideally, capital, means that the law of variable proportion is associated with the short-run production function. As a result, we assume the production function Q = f ($\overline{K}$, L), where $ \overline{K}$ is a fixed factor in the short run.

## Assumptions of Theory

- Technology remains fixed.
- At least one factor (capital in our case) must be constant.
- Factor inputs (as we have assumed labor and capital) are not perfect substitutes.
- Variable factor (labor) is homogeneous.

## Statement of Theory

The theory states that when the variable factor (labor) is employed more and more with the given fixed factor (capital), the total product of labor (TPL) increases at the increasing rate at the early stage of production, after a certain point total product of labor increases at the decreasing rate, reaches maximum and starts to decline. In brief, when more units of variable factor are successively used with the fixed factor, the marginal product of labor eventually decreases. This is why, this theory is also known as the law of diminishing returns (or diminishing marginal productivity). This way, the law of variable proportion shows that how output changes when more and more variable inputs are used with a constant amount of fixed factor.

## Explanation of theory

Let us explain the theory with a numerical example. Given the fixed 5 units of capital, suppose the number of labors and corresponding outputs as follows:

In the table given above, the total product of labor increases at the increasing rate up to the 3rd unit of labor because the marginal product of labor increases up to the 3rd unit. After 3rd unit, the total product of labor increases at a decreasing rate because the marginal product of labor has started to decline thereafter.

However, the average product continues to rise and reaches a maximum when increasing the average product and decreasing the marginal product equal at 4th unit. After 3th unit, the marginal product decreases at a faster rate than the average product increases. Hence, the average product also starts to decline after 4th unit but never touches the horizontal axis.

Likewise, the marginal product of labor continuously decreases to become zero at 6th unit at which the total product reaches the maximum. After that, a marginal product becomes negative after 6th unit. And, accordingly, the total product starts to decline reaching its maximum at 6th unit. Also observe that the K/L ratio changes as the labor changes given the fixed capital, hence, the name of this theory is a **law of variable proportion**.

**Pictorial presentation**

This numerical value in the table can be shown graphically as follows:

Initially, the shape of the TPL curve is convex to the origin. The convex shape results from the negative marginal productivity of fixed factor (capital) (why?) which turns out to be positive with the usages of more and more variable factors. After a certain period, the TPL curve is concave to the origin because of the diminishing marginal product of labor. There are three stages of production- first, second and third. We will explain each of them as follows:

**First stage**:

This stage of production is the stage of increasing returns because the average product increases and reaches the maximum when the marginal product equals the average product despite the marginal product has already begun to decline. Hence, the first stage of production goes from zero units to 4th i.e. L2 units of labor. The first stage ends with the start of the fall of APL. The main causes of the increasing returns are the fuller and more efficient utilization of fixed factors and specialization of labor.

**Second stage**

This stage of production is diminishing returns because both average product and marginal product decline but remain positive throughout this stage. Thus, the second stage starts with the fall of the average product and terminates when the total product reaches maximum (when MPL is zero). In the figure presented, this second stage lies between L2 to L3 units of labor. At this stage, output increases at a decreasing rate because of decreasing marginal product. Nonetheless, APL keeps on increasing until marginal product equals average product.

The main causes of diminishing returns are the excess of a variable factor over the fixed factor, the indivisibility of the fixed factor, and no perfect substitution between factor inputs. Prof. Bober rightly remarks on indivisibility: “Let the indivisibility enter through the door, the law of variable proportion rushes out through the window.” In addition, a scarce factor is taken as the fixed factor and its supply cannot be increased. And, also the factors are not perfect substitutes by assumption.

**Third stage**:

This is the stage of negative returns because of the negative marginal product of labor. The total product declines in this stage due to the negative marginal product of labor. However, the average product remains positive so long as the total product is positive. The main causes of the negative returns are excess of variable factor (labor) over the fixed factor (capital) and difficulty in management and control. This situation is rightly underscored by the proverb – “too many cooks spoil the broth”.

## Stage of operation for a rational producer

A rational producer never produces at the third stage because in this stage marginal product is negative and reduces the total product. Cutting the number of labor would provide the producer the more output. Hence, the third stage is not the stage of operation for a rational producer. Similarly, a rational producer does not produce at the first stage because at this stage marginal product of the fixed factor (capital) is negative. Increasing the variable factor means the fuller and more efficient use of fixed factor. Hence, the producer does not produce in the first stage. The only remaining stage of operation is the second stage.

A rational producer operates in this stage because the total output increases in this stage despite decreasing average product and marginal product. Hence, a rational producer operates somewhere between the 4th and 6th units. But, the exact point at which the producer operates depends on the relative price of the factors.

## Application of the law of variable proportion

Initially, the law of variable proportion is considered to operate in agriculture production only. However, this law has vast and universal applicability and applies in both the agriculture and industry sectors as well. Moreover, application of diminishing returns means that the future of mankind looms large as a gloomy picture. There might happen starvation, underproduction, shortage, hardships, misery, and so on due to continual productivity decline.

Nonetheless, why we are not observing these dismal situations universally in the real world? The answer lies within the assumption of ‘other things remaining the same’. That means we have assumed the technique of production to remain constant throughout the analysis of theory. However, technological advancement is a continual process and keeps on developing, which ultimately contributes to higher productivity growth.

In a nutshell, we can suspend the operation of diminishing returns through the continual advancement in production techniques.