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Correct answer is (b) maths

Law of Diminishing Returns states that when more and more units of variable factors are employed with a fixed factor, then marginal product of the variable factor must fall.

Product or output refers to the volume of goods produced by a firm or an industry during a specified period of time.

The required solution is:

Production Function is an expression of the technological relation between physical inputs and output of a good.

Production function is the physical or quantitative relation between production and the means which contribute in production. 

Following are the definitions of production function-

1. In words of Henderson and Quandt, “Production function is an engineering concept which explains the relationship between input and output. This relation is quantitave and assumes a fixed, given technology”.

2. As per Dr Balwant Kandoi, “If quantity of production is Q and the factors of production like labour, capital, land, management and technique (Ld, L, K, O) are used, then we will write the Production function as:
Y = f (Ld, L, K, O)”

3. According to the N Gregory Mankiw, “Relationship between physical quantity factors of production or inputs and the production is production function”.

4. According to Watson, “The relationship between a firm’s physical production (output) and the material factors of production (inputs) necessary to produce it is called production function”.

On the basis of the above definitions, production function can be defined in simple terms as follows :

Production Function is the physical relation between Input and Output.

General characteristics of Production Function

1. Production Function is cost independent- Production Function has no relation with the price of produced goods and the cost of means used to produce them. Its relation is mainly with the physical quantities of production. Although, each producer, at the time of deciding the level of production or selecting the combination of means, always keeps the price of goods and means in his mind.

2. Production Function is an Engineering concept, not an economic concept- The task of Production Function is to interpret the relation between the means of production and the physical level of production. Therefore, it is considered as an Engineering concept, not on economic concept.

3. Production Function is a time dependent concept- The relation of production function is with a specific time period. It does not have any meaning if the time period is changed. That is why it is considered as a time dependent concept.

4. Production Function accepts the substitution possibilities of means – The concept of Production function accepts that the substitution possibilities of means of production exist, meaning one means can be used in place of another. Example- capital can be used in place of labour.

5. Production Function accepts the complete divisibility of means- Production Function accepts the complete divisibility of means; that means, it accepts that every mean of production can be divided.

6. Production Function is related to a definite given technique- The technique of production must not vary, it is based on the single technique. When the technique is of lower grade, without even decreasing the level of means, the production can be decreased.

7. Production Function is the subject of static economics- The concept of Production Function is the subject of static economics because under this, cost of means, level of technical knowledge and the timeline is considered to be constant.

8. Production Function is short-term or long-term on the basis of time period- Production Function is categorised as short-term or long-term on the basis of time period. There is a difference between these two types of production functions on the basis of proportion of means. In short-term, the ratio of varied and constant means keeps on changing with the production, while in long term, the ratio of every means remains equal.

Ip the long-term production function, every means is variable, none of them is static. Therefore, there is no straight line drawn above the means in the formula of long-term production function. The formula of production function in this case is – P = f (L, Ld, K, T, E)

In the short-term production function, only labour is a variable factor. Rest means are constant. Therefore, other than the labour, every means is highlighted with a line drawn above them.

Today, economists have started considering the law of returns of variable proportions and law of diminishing marginal production as one. According to the law of returns of variable proportions, with the use of fixed means along with the variable menas, the ratio of means get changed and its effect is visible on the production. This effect can be seen on the increasing returns, equal returns and diminishing returns, and the condition of increasing returns is in the initial stage only. Eventually, it means diminishing returns only. Therefore, both these laws are considered to be the same.

In conditions of returns to scale, the ratios of combination of factors of production remain constant as earlier. But in returns to outlays, the combination proportions change.

The Production function is only related to the physical quantities of the production. It has no relation with the costs of means of production or the produced goods. But still, this is also true that, when each producer decides the level of production and selects the combination of means, he always keep the price of goods and means in his mind.

Production function is considered as a subject of static economics because in the theories of Production function, the cost of means of production, the level of technical knowledge and time period is considered static. That is why, it is not a part of dynamic economics, but of static economics.

The concept of production function accepts that the substitution of means of production is possible, meaning that one factor can be used in place of means another. 

Example- capital can be used in place of labour.

It is one of the main assumptions of production function, that the technique of production should not be varied. This is because, the production of goods is affected by the level of factors, along with the dependence on production technique. Better techniques result in better production of goods. When the technique is of lower grade, without even decreasing the level of factors, the production can be decreased. And that is why, with a definitive technique of production function, it can be said that what will be the effect on the level of production with the variation in level of production means.

It is related to short time period.

The following are the assumptions of production function :

1. There should be no change in the level of technical knowledge.
2. The costs of means of production should remain the same.
3. Production function is related to a definite time period.
4. The means of production should be divisible.
5. The most efficient techniques of production should be employed.
6. The means of production can be substituted upto a limited extent.
7. There is uniformity among the means of production.
8. The objective of the firm should be maximization of production.

In the case of returns to scale, changes are made in all sources by equal proportion or percentage, not by different proportions or percentages. As such, changes are not made in land area by 20%, capital by 10% and workers by 50%. When the means of production are converted according to the expenses incurred on them in equal or varying proportion, then it is known as the Return of Outlay/Expenditure.

French Economist Prof. Turgot.

(b) every factor is varied

(d) Both (a) and (b)

(b) Factors below the straight line are constant

(a) Inputs and outputs

In short-term, the proportion of fixed and variable factors keeps on changing along with the change in production. On the contrary, in the long-term, all factor inputs can be varied,

Production function are of two types:

1. Cobb-Douglas production function
2. Linear-Homogenous production function.

In economics, to make decisions for optimum production, the information about different alternative production functions of any goods and services is necessary.

The means through which production is done is called Input in economics.

Production Function can mathematically be stated through the following equation:
X = f (a, b, c….n)
Where, X = production level of the goods, a, b, c, etc. are the means of production like land, labour, capital, entrepreneurship, etc.
f = function.

The commodity that is produced by a firm is known as Output in economics.

The concept of Production Function is the subject of a static economics because under this, cost of factors, level of technical knowledge and the time period is taken to be constant.

Two definitions of Production Function are:

1. In words of Henderson and Quandt, “Production function is an engineering concept which explains the technological relationship between physical inputs and outputs of the goods present in production”.
2. According to the N Gregory Mankiw, “Relationship between factors of production and production is production function”.

When one production factor is changed while keeping all other factors constant, it is called short-term production function.

(c) some are constant and some are varied

The functional relationship between physical input and physical output of a firm with a given state of technology is known as Production Function. A production function is a mathematical statement used to describe the technological relationship between input and output in physical terms. In its general form, it tells that production of a commodity depends on certain specific inputs. It presents the quantitative relationship between input and output. Besides, the production function represents the technology of a firm, of an industry, and of the economy as a whole. A production function may take the form of a schedule or table, a graphed line or curve, an algebraic equation or a mathematical model. But each of these forms of a production function can be converted into its other forms.

Definition of Production Function
The concept of production has been differently defined by different economists as stated below:

1. In words of Alfa. C. Chiang, “Function is a group of combination for variables (independent and dependent) in a specific order. The specialty of this function is that any value of X determines a unique value of Y.”

2. In words of Henderson and Quandt, “Production function is an engineering concept which explains the relationship between input and output. This relation is quantitave and assumes unchanging technology”.

3. As per Dr Balwant Kandoi, “If quantity of production is Q and the factors of production like labour, capital, land, management and technique (Ld, L, K, O, T) are used, then we will write the Production function as:
Y = f (Ld, L,K,0, T)”

4. According to the N Gregory Mankiw, “Relationship between factors of production or inputs and outputs is production function”.

5. According to Watson, “The production function is the relationship between a firm’s physical production (output) and the material factors of production (inputs) necessary to produce it.”

Features of Production Function

Production function has the following main features:

1. It indicates a functional relationship between physical input and physical output of a firm.
2. The production function is always in context to a period of time.
3. The production function can specify either the maximum output that can be produced from a given set of inputs or the minimum quantity of inputs required to produce a given level of output.
4. Production function is purely a technical relationship. It has no reference to cost of inputs or costs of products.
5. Output in the production function is the result of combined use of factors of production.
6. The production function includes all the technically efficient methods of production.

Assumptions of Production Function

1. It presumes a fixed given technology.
2. The costs of means of production remain the same.
3. Production function presumes a definite time period.
4. Combinations of factors of productions can be changed to an extent only.
5. Factors of production are homogenous, i.e., labour = labour and capital = capital.
6. Factors of production are variable.
7. Substitutability of factors of production is upto a limited extent.
8. The objective of firm is output maximization.

When factors of production are changed by different ratios or percentages on the basis of the overheads/expenses (outlays) ‘done on them, then it is called “Returns to Outlays”. Proportion or ratio of factor changes when outlays on factors of production are changed in an equal proportion or ratio.

Function is a technical term of mathematics. It means the quantitative relationship between two variables (independent and dependent variable). Like y = f (x), in which y is a dependent variable, that depends on the independent variable x. It means that the value of y is dependent upon the value of x. It also means that y is that dependent variable which is mathematically related to the independent variable x.

Following are the main characteristics of production function:

1. Production function is an engineering concept.
2. Production fundction is cost independent, i.e. it is not dependent on costs.
3. Production function is related to the flow of means and production.
4. On the basis of time period, it is short-term and long-term.
5. Production function relates to a fixed time factor.
6. It is related with the given technology.
7. Production function accepts the substitution possibilities of means of production.

The production of any commodity depends on the means of production. Thus, the relationship between the means of production and its production is called production function. It defines the physical relationships between quantity of production and means of production in a given time period. It is a quantitative relationship, not a qualitative one.

In economics, return of scale is established on that situation in which all the means of production are converted into a certain ratio or percentage.

This production function was rendered by Prof. C. W. Cobb and P.H.Douglas.

Scale can be a particular unit of measurement, like metre, litre, kilogram, hectare, etc.

Function means the quantitative relationship found between two variables (independent and dependent variables).

1. It indicates a functional relationship between physical input and physical output of a firm.
2. The production function is always applied in context to a certain period of time.

Scale can imply any one specific unit of measurement, like metre, litre, kilogram, hectare, etc.

Production function expresses the technical relationship between production inputs and the final product. It shows the fundamental relationship between quantity of production and means of production for a given time period.

The Cobb-Douglas production function is considered to be the most important.

1. A fixed, given technology is used.
2. Costs of means of production remain the same.

It is related to long time period.

Inputs imply the means of production.

P = F (L, K, Ld, M, T, C)
Where,
P = production function
F = Function
L = Labour
C = Capital
Ld = Land
M = Management
T = Technology
C = Courage or Entrepreneurship.

In words of Alfa. C. Chiang, “Function is a group of combination for variables (independent and dependent) in a specific order. The specialty of this function is that any value of X determines a unique value of Y.”