Production Function || Law of Variable Proportions

Production Function:

Production is the process through which certain commodities and services, known as inputs, are turned into other goods and services, known as outputs. The link between the input of factor services and the output of the final product is referred to as the production function. The production function is founded on the assumption that the amount of output in a manufacturing process is proportional to the number of inputs utilised in the process.

Output is dependent on input or a group of inputs in such a manner that each combination of inputs results in a unique quantity of output. The production function refers to this one-of-a-kind link between output and input.

1.  Linear Homogeneous Production Function:

The production function is considered to be homogenous when all of the inputs are raised in the same proportion. The degree of the production function is one. This is referred to as a linear homogeneous production function. To estimate the production function, the function must be expressed in explicit functional form. This type of production function may be described mathematically as

nQ = f (nL, nK)

This production function also implies that the returns to scaling are constant. That is, if L and K are multiplied by n, the output Q is multiplied by n as well. A well-behaved production function is this type of production function. Which makes the entrepreneur's job much easier and more convenient? He merely needs to determine one of the best factor proportions.

2.  Cobb-Douglas Production Function:

Charles W. Cobb and Paul H. Douglas investigated the link between inputs and outputs and developed an empirical production function known as the Cobb-Douglas production function.

Originally, the C-D production function applied not to a single firm's production process, but to the entire industrial production.

The Cobb-Douglas production function is denoted by

Q = ALαKβ

where Q is the output and L and A' are the labour and capital inputs, respectively. A, and are positive parameters with > 0, > 0. The equation states that output is closely related to L and K and that the part of the output that cannot be described by L and K is explained by A, which is the residual, also known as technical change.

The marginal products of labour and capital are functions of the parameters A, and, as well as the labour and capital input ratios. In other words,

MPL =∂Q/∂L = αAL α-1K β

MPK = ∂Q/∂K = βAL αK β-1

3.  Constant Elasticity of Substitution Production Function:

The Homohighplagic production function is another name for the CES production function. The Constant Elasticity of Substitution (CES) function was created by Arrow, Chenery, Minhas, and Solow. This function has three variables, Q, K, and L, as well as three arguments, A, a, and 0. It can be represented in the following way:

Q = A [α C-ϴ+ (1- α) L -ϴ]-1/ϴ

When Q denotes total production, K denotes capital, and L is labour. A is the efficiency metric that indicates the status of technology and production organisational features. It demonstrates that as technological and/or organisational changes occur, the efficiency parameter shifts the production function, an (alpha) is the distribution parameter or capital intensity factor coefficient concerned with the relative factor shares in total output, and 0 (theta) is the substitution parameter determining the elasticity of substitution. And 

A > 0; 0 < α <1; ϴ > -1.

The elasticity substitution in the CES production function is constant and does not have to be equal to unity.

Mukherji created the CES function by combining more than two inputs.

4.  Variable Elasticity Substitution Production Function:

Bruno, Knox Lovell, and Revankar have recently attempted to get a new manufacturing function. The resultant production function is an extension of CES with the desirable variable elasticity substitution features.

To describe value added per unit of labour, Lu and Fletcher constructed a logarithmic connection using the wage rate (W) and the capital-labour ratio (K/L).

V/L = a + b log W + с log K/L where

V = Value added,

W = Wage rate

K = Capital,

L = Labour 

a, b and с are the parameters to be estimated.

The elasticity of substitution (σ) is σ = b/1-c (1+WL/rk)

where WL and rk are the shares of labour and capital respectively.

L aw of Variable Proportions

Up to Alfred Marshall's time, three rules of returns were referenced in the history of economic philosophy. These were the laws of growing returns, decreasing returns, and constant returns, respectively. According to Dr Marshall, the law of declining returns applies to agriculture and the law of growing returns to industry. Much time was spent debating this subject. It was eventually discovered, however, that there are no three rules of production.

It is just one production rule with three phases: growing, decreasing, and negative production. The Law of Variable Proportions or the Law of Non-Proportional Returns was called after this basic law of production.

Assumptions:

Under the following assumptions, the law of varying proportions, often known as the law of decreasing returns, remains true:

(i)  Short-run

(ii)  Constant technology

(iii)  Homogeneous factors

Three Stages of the Law:

According to the law of varying proportions, there are three steps or phases of production:

(i)  Increasing returns.

(ii)  Diminishing returns.

(iii)  Negative returns.

Diagram/Graph:

These stages are illustrated in the graph below:

(i)  Stage of Increasing Returns: The first step of the law of changing proportions is commonly referred to as the growing returns stage. As a variable resource (labour) is added to fixed inputs of other resources at this stage, the total product rises up to a point at a rising rate, as seen in figure 11.1.

The total product grows at an increasing rate from the origin to point K on the slope of the total product curve. The total product continues to rise from point K onwards throughout stage II, although its slope is decreasing. This indicates that starting at point K, the total product grows at a decreasing pace.

Causes of Initial Increasing Returns:

When the amount of a constant element is abundant relative to the quantity of a variable factor, the period of rising returns begins. It is employed more intensely and efficiently as more units of the variable component are added to the constant quantity of the fixed factor. As a result, manufacturing increases at a quick pace. Another reason for growing returns is that the original fixed component is indivisible. As more units of the variable factor are used to work on it, production grows dramatically as the variable component is used more fully and effectively.

(ii)   Stage of Diminishing Returns: This is the most crucial step of the manufacturing process. In stage 2, overall production increases at a decreasing pace until it reaches its maximum point (H), at which time the second stage concludes. Both the marginal product (MP) and the average product (AP) of the variable factor are decreasing but still positive at this point.

Causes of Diminishing Returns:

When the fixed component becomes insufficient in relation to the quantity of the variable factor, the rule enters the second phase. The marginal and average product decrease when more units of a variable component are used. Another cause of decreasing returns in the production function is that the fixed indivisible factor gets overworked. It is being used in a non-optimal situation! Mrs J. Robinson goes a step further, claiming that decreasing returns arise because the factors of production are unsatisfactory replacements for one another.

(iii) Stage of Negative Returns: Total production decreases in the third stage. The TP curve is slanted downward (From point H onward). The MP curve reaches zero at L2 and then turns negative. As the utilisation of variable factors increases, it moves below the X-axis (labour).

Causes of Negative Returns:

The third phase of the rule begins when the number of a variable, factor becomes excessive in comparison to the fixed factors. A producer cannot function at this level since the utilisation of more labour reduces total output.

A sensible producer will always want to create in stage 2 when the variable factor's MP and AP are decreasing. The price of the element that the producer must pay determines when he will decide to produce. The variable factor (say, labour) will be used by the producer until the marginal product of labour equals the stated wage rate in the labour market.