Returns to a factor means the increase in the Total Product (TP) per unit increase in the employment l of a variable factor, when the employment of other factors is kept constant. 
Increasing Marginal Product leads to increasing returns to factor. In other words, when Total Product  increases at an increasing rate, increasing returns to factor takes place. This can be explained with  the help of following table.

In the table, MarginalProduct increases by employing an additional unit of variable factor. That is,Total Product is increasing at an increasing rate.Thus increasing MP leads to increasing returns of factor

The Law of variable Proportions explains the behaviour of Total Product ~nd Marginal Product. It states that when more and more units of a variable factor are employed, keeping the other factors fixed, the Total Product (TP) first increases at an increasing rate, then increases at a diminishing rate and finally starts falling.The different phases in the behaviour of TP and MP can be explained as below.

Average product is defined as the output produced by per unit of variable factor (labour) employed. Algebraically, it is defined as the ratio of the total product by units of labour employed to produce the output, i.e. 
\(\mathrm{AP}=\frac{\mathrm{TP}}{L}\)
Where,
TP = Total product
L = units of labour employed

Total product is defined as the sum total of output produced by a firm by employing a particular input. It is also known as the Total Physical Product and is represented as
\(\mathrm{TP}=\sum Q_{x}\)
Where, ∑ represents summation of all outputs and Q_x represents units of output x produced by an input.

The production function of a firm is a relationship between inputs used and output produced by the firm. For various quantities of inputs used, it gives the maximum quantity of output that can be produced. Ir other words, the production function shows the technical relation between factor input and output.
The inputs that a firm uses in the production process are called factors of production. Suppose a firm produces output using only two factors of production, factor I and factor 2. The production function, hence, shows the maximum quantity of output that can be produced by using different combinations of these two factors. Thus, a production function can be expressed as:
 \(q=F(x_{ 1 },x_{ 2 })\)
According to the above equation, the firm can produce at most q units of output by using \(x_{ 1 }\) amount  of factor I and \(x_{ 2 }\)  amount of factor 2.

Decreasing returns to factor implies that keeping other inputs constant, an increase in the variable input will cause the Marginal Product to decrease after a certain level of output. It is also known as the Law of Diminishing Marginal Returns.
Following are the causes of Diminishing Marginal Returns to Factor:
(i) Fixity of Factors: When more and more units of variablefactor are combined with the fixed factor, the latter gets over-utilised. As a result, the Marginal Product from each additional unit of variable factor declines.
(ii) Imperfect Substitution: Factors of production are imperfect substitutes of each other. It is possible to use one factor as a substitute for another factor of production only till a limit. However, production beyond that limit causes diminishing marginal returns.
(iii) Scarcity of Factors of Production: Every factor of production is available in limited quantity. Availability of limited factor decreases asthe level of output is increases. Consequently, the cost increases and marginal product starts to decline.

The factors of production can be classified as fixed and variable factors. The fixed factors cannot be changed over a short period. The Law of Variable Proportions (or the Law of Returns to a Factor) states that when more and more units of a variable factor are employed, keeping the other factors  fixed, the Total Product (TP) first increases at an increasing rate, then increases at a diminishing rate and finalIy starts falling.The law explains the change in output due to the change in the proportions between the fixed and variable factors. The law of returns to a factor can be explained with the help of following diagram:
Stage I: Increasing Returns to Variable Factor
The Total Product (TP) increases at an increasing rate from 0 to L, units of labour. That is, Marginal Product (MP) is increasing.
Stage 2: Diminishing Returns to Variable Factor
The Total Product (TP) increases at a diminishing rate from L, to L, units of labour. That is, Marginal Product (MP) is decreasing. MP is 0 at ~ units of labour. •
Stage 3: Negative Returns to Variable Factor
The Total Product (TP) is decreasing beyond ~ units of labour. The output decreases as an additional unit of labour is employed. That is, Marginal Product (MP) is negative.

Relationship between marginal products (MP) and the total product (TP) can be represented graphically as

1) TP increases at an increasing rate till point K, when more and more units of labour are employed. The point K is known as the point of inflexion. At this point MP (second part of the figure) attains its maximum value at point U.
2) After point K, TP increases but at a decreasing rate. Simultaneously, MP starts falling after reaching its maximum level at point U.
3) When TP curve reaches its maximum and becomes constant at point B, MP becomes zero.
4) When TP starts falling after B, MP becomes negative.
5) MP is derived from TP by
\(\mathrm{MP}=\frac{\Delta \mathrm{TP}}{\Delta L}\)
Or, MP = TP_n - TP_n-1

Marginal Product is defined as the additional output produced because of the employment of an additional unit of labour. In other words, it is the change in the total output brought by employing one additional unit of labour. Algebraically, it is expressed as the ratio of the change in the total product to the change in the units of labour employed, i.e.
  \(MP=\frac { \Delta TP }{ \Delta { X }_{ 1 } } \) 
or, MP_L = TP_n - TP_n-1
Where,
TP_n = Total product produced by employing n units of labour
TP_n−1 = Total product produced by employing (n − 1) units of labour.

Increasing returns to scale implies that a proportionate change in the output is greater than the proportionate change in input. For example, the output increases by 20 percent due to 15 percent increase in all the inputs. Increasing returns to scale are due to economies of scale. Economies of scale indica e that long run Average Cost decreases, which corresponds to increasing returns to scale in terms 0 output. Economies of scale can be classified as::
1. Internal Economies of Scale: These include
(i) Technical Economies
(ii) Financial Economies
(iii) Marketing Economies
(iv) Managerial Economies
(v) Labour Economies
(vi) Risk-bearing Economies
(vii) Economies due to Indivisibilities
2. External Economies of Scale:These include
(i) Economies of Concentration
(ii) Economies of Information
(iii) Economies of Disintegration

(i) False. Increasing in Total Product indicates increasing returns to a factor as well as decreasing returns to a factor. When Marginal Product increases, Total Product increases at increasing rate causing increasing returns to a factor. When Marginal Product decreases, Total Product increases at decreasing rate causing decreasing returns to a factor.
(ii) False. Marginal Revenue is not always the price at which the last unit of commodity is sold. It is rue only under perfectly competitive market structure. Marginal Revenue can be zero or even negative under other forms of market, but-price cannot be zero or negative
(iii) False. When there are diminishing returns to a factor, only Marginal Product falls. However, Total Product increases but at a diminishing rate. Total Product tends to increase as long as Marginal Product is positive. When there are diminishing returns to a factor marginal and total products both always fall is a false statement. We can understand this situation with the help of schedule.