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BCom 1st Year Production Function Notes Study Material

BCom 1st Year Production Function Notes Study Material

BCom 1st Year Production Function Notes Study Material

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BCom 1st Year Production Function Notes Study Material
BCom 1st Year Production Function Notes Study Material

BCom 1st Year Production Function Notes Study Material


The laws of production describe the technically possible ways of increasing the level of production. Production means the transformation of inputs into outputs. Inputs are the things bought by a firm. Outputs are the things the firm sells.

The words ‘inputs’ and ‘factors of production’ are near synonyms and in many contexts are used interchangeably. In general, however, the connotation of inputs is broader. Inputs are all the things that firms buy. Factors of production take on a narrower meaning and mean labour and capital generally. A synonym for factors of production is “productive services”. The words “output”, “product” and “production” are exact synonyms.

Production Function

A production function is an economic summary or description of technological possibilities; it is the framework the economist uses to record what he needs to know about the physical aspects of the production process.

Although technology (methods of combining resources) may change rapidly, at any particular time it can be taken as given. The producer surveys production possibility within the limits of available technical knowledge. A production function can be presented as a schedule or as an algebraic function with output dependent on possible input combinations.

In the second case, production function describes the relation between physical inputs and physical outputs of a firm. The laws of production are a study of the production function. Algebraically, production function can be written as:

Q = f (A, B, C, D,…)

where, Q = quantity of output

A, B, C, D,…. = quantities of the factors a, b, c, d,….

Knowledge of production function is a technological or engineering knowledge.

Economic theory looks to two kinds of input-output relations in production function.

(a) There are situations where it is not possible or feasible to expand all inputs in the same proportions, according to scale. A special case occurs when one or several inputs remain fixed, while one or several inputs are varied with respect to fixed factors, such as, land, plant, equipment or managerial capacity. Which factors are fixed and which are variable really depend on the problem being analysed. The Marshallian time periods, which bear no direct or ncessary relationship to clock time, are useful concepts in the economic analysis of problems.

(b) In the long-run, the problem permits expanding all inputs according to scale so that no one input is necessarily more variable than another. Most often this situation, where all factors are variable, applies to the problem of planning the optimal size of the enterprise. In the short-run, some inputs are fixed and some (or one) are variable.

The major forces in the short-run are on the effects of changing factors proportions. The long-run situation involves considerable planning and capital investment which often cannot be quickly reversed or modified.

Short-run Production Function: The Law of Variable Proportions

The costs of an individual firm are affected by two sets of considerations, namely, (i) the proportions in which the firm combines factors and (ii) the scale on which it operates. The law of variable proportions deal with the first set and it is assumed that scale has no influence.

“As equal increments of one input are added”, says Stigler, “the inputs of other productive services being held constant, beyond a certain point, the resulting increments of product will decrease, i.e., the marginal products will diminish.”

When the quantity of one factor is varied, keeping the quantity of other factors constant, the proportions between the variable factor and the fixed factor are altered. Hence it is called the law of variable proportions. It is also known as the law of non-proportional output. It is the law of diminishing returns of the classical economics which was stated by Marshall in the following words:

“An increase in the capital and labour applied in the cultivation of land causes in general a less than proportionate increase in the amount of produce raised, unless it happens to coincide with an improvement in the arts of agriculture.”

In the statement of this law. Marshall takes land as the fixed factor, while labour and capital are variable factors. The operation of the law is confined to agriculture in this definition. The modern version of this law in terms of variable proportions suffers from no such limitation and is applicable to the case of any factor whose supply is fixed in the short-run.

A firm’s costs of producing any output depends not only upon the prices of needed resources, but also upon technology—the quantity of resources it takes to produce that output. In the short-run a firm can change its output by adding variable resources to a fixed plant.

Here the question that arises is how does output change as more and more variable resources are added to the firm’s fixed resources. The answer is provided in general terms by the law of diminishing returns (i.e., the law of variable proportions). This is an engineering law which states that as successive units of a variable resource (say, labour) are added to a fixed resource (say, capital), beyond some point the marginal product attributable to each additional unit of the variable resource will decline. When the law is stated in this way, there is no attempt to confine its operation to any particular area agriculture or non-agricultural industries. It holds true in both areas.

Assumptions of the Law

The law of variable proportions holds true under the following assumptions:

(i) The state of technology is given. The various rates of input of the productive services are simultaneously available alternatives, and not a historical sequence. This must be emphasized because for many decades after economists first adopted this law in 1815, its chief application was to the growth of population relative to land. Thus it was believed that diminishing returns would dominate technical progress. Economists no longer have this belief, and so hold to constant state of technology.

(ii) It is necessary that there are productive services whose quantity is held constant. This law does not apply when all inputs are harmoniously varied, at least in quite so simple a form; this is a problem of the economies of scale. The law premises the possibility of varying the proportions in which the various productive services combine.

Obviously, if two productive services must be used in rigidly fixed proportions, an increase in one without a proportionate increase of the other will lead to no increase of output (marginal product will be zero, and not diminishing). For the law to operate, it is necessary that the proportions in which productive services combine generally be variable. The law is important, and rigidly fixed proportions are quite uncommon.

Explanation of the Law

When one input is variable, the relation between the input and product are conventionally divided into three ‘stages’. In order to explain the three stages, we take the help of Table and fig.

Explanation of the Law

It has been shown in the table that a farmer employs one labourer on one acre of land to produce, say, wheat. The total product from this piece of land is 100 kg of wheat. The average product and marginal product are the same in this case. When two labourers are employed, the land is utilised more effectively. Consequently, total product rises at an increasing rate and reaches 220 kg. The marginal product of the second labourer increases to 120 kg and the average product of two workers goes up to 110 kg.

Two labourers fully utilise that piece of land. So further increase in labourers applied to this plot of land results in the rise of total product but at a diminishing rate. Average product and marginal product begin to decline from the third unit of labour and continue to do so till the sixth unit of labour is employed. Total product is highest at this point. Seventh labour adds nothing to the total product, so its marginal product is zero. Addition of the eighth labour causes the total product to decline and the marginal product becomes negative. The three stages have been presented in fig.

Explanation of the Law

It can be seen in the figure that the total product curve rises at an increasing rate in the beginning and then ascends at a diminishing rate. It reaches its highest point G and then begins to decline. The slope of TP curve is ∆TP/∆L, where L stands for labour. The slope of TP curve expresses marginal product. This slope is continuously changing. At point E its slope is the maximum because just below it at H, highest marginal product is obtained. E is known as the point of inflection. At point G, the slope of TP curve is zero because here the marginal product reaches zero (at B).

In stage I, marginal product increases first and then begins to decline. The boundary between stage I and stage II corresponds to the point where the average product is a maximum.

In stage II, marginal product continuously declines. The boundary between stage II and stage III is similarly marked by the maximum point on the total product curve, beyond which point the marginal product becomes negative.

In stage III, marginal product is negative and total product declines. It is an irrational stage as no producer can ever think to operate in this stage.

Stage I is called the stage of increasing returns and stage II the stage of diminishing returns, while stage III is the stage of negative returns. In stage I, the marginal product of the fixed factor (here, land) is negative, while in stage III marginal product of the variable factor (here, labour) is negative.

Three Stages of the Law of Variable Proportions: Explanation

(i) Stage I

In this stage the fixed factor is abundant relative to the variable factor. So when more and more units of the variable factor are added, the fixed factor is more intensively and effectively utilized. As a result, the marginal product of the variable factor increases in the beginning.

At this point a pertinent question is raised. Why is the fixed factor not initially taken in a quantity which suits the available quantity of the variable factor? The answer is the “indivisibility” of the factor.

Indivisibility: It is the characteristic of a factor of production or commodity which prevents its use below a certain minimum level. Many types of plant and machinery have, for engineering reasons, a single most efficient size. Consequently, it will be either technically impossible to make the equipment of a different size, or the costs of production associated with other sizes are higher.

So as the scale of output increases up to this optimum, increasing productive efficiency is achieved. “The word ‘indivisibilities’ is used to categorize the sources of scale economies because they would not arise if the plant and processes were capable of being increased or decreased in scale by small amounts without any change in their nature, i.e., if they were perfectly divisible.” (Penguin Dictionary of Economics)

Because of the indivisibility of a factor, a minimum amount of that factor has to be employed whatever the level of output. When more units of the variable factor are employed to work with an indivisible fixed factor, output generally increases due to fuller and more effective utilisation of the fixed factor.

There is yet another explanation of increasing returns in stage I. As more units of the variable factor are employed, the efficiency of the variable factor itself increases. The reason for this is that when the variable factor is used in sufficient amount, the possibility of introducing specialisation and division of labour is increased.

(ii) Stage II

It needs to be explained why diminishing returns appear in stage II.

Increasing returns appear in the first stage primarily because of the more effective and efficient use of the fixed factor as more and more units of the variable factor are combined to work with it (the fixed factor),

Now once the point is reached at which the amount of the variable factor is sufficient to ensure the efficient utilization of the fixed factor, any further increases in the variable factor will cause the marginal and average products to decline. It will happen so because the fixed factor then becomes inadequate relative to the quantity of the variable factor.

In stage I, fixed factor is abundant relative to the units of the variable factor. In stage II, the fixed factor becomes scarce in relation to the variable factor. As a result, the increase in the units of the variable factor receives less and less aid from the fixed factor. So decline sets in average and marginal products.

The phenomenon of diminishing returns, like that of the increasing returns, rests upon the indivisibility of the fixed factor. A certain amount of the variable factor is necessary for the most efficient and fuller utilization of the indivisible factor. When that point is reached, further use of the variable factor means less efficient use of the indivisible factor.

Mrs. Joan Robinson is of the opinion that diminishing returns appear because the factors of production are imperfect substitutes for one another. When fixed factor becomes scarce, had there been perfect substitute for this factor, there would be no diminishing returns. In her words,

“What the Law of Diminishing Returns really states is that there is a limit to the extent to which one factor of production can be substituted for another or, in other words, that the elasticity of substitution between factors is not infinite. If this were not true it would be possible, when one factor of production is fixed in amount and the rest are in perfectly elastic supply, to produce part of output with the aid of the fixed factor, and then when the optimum proportion between this and other factors was attained, to substitute some other factor for it and to increase output at constant cost.”

(iii) Stage III

In this stage, variable factor becomes abundant relative to the fixed factor. So even if some units of the variable factor are taken away, there would be no effect on production, or perhaps, total product will increase. Hence marginal product takes negative value. Although stage III is an irrational area of production, it is not uncommon for firms, lacking perfect knowledge, actually to produce in that region, particularly in agriculture. As Bishop and Toussaint note:

“……….evidence of production in stage III is often noted. For example during the late summer and fall months, we frequently have evidence of too many cattle on a given quantity of pasture, resulting in overgrazing of pastures and less production than could have been obtained with fewer cattle. Also, we find evidence of overcrowding of broilers and layers in poultry houses.” (Breit and Hochman (Ed.), Readings in Micro Economics)

Economic Region of Operation

In the short-run, a rational producer would not choose to remain in stage III; if he were there by some error in calculation, he could increase his to product by reducing his use of the variable factor. Thus in stage III, he would be overutilizing his fixed resources, and some of his variable inputs would be economically redundant.

Another way of putting it would be to state that even if the variable resource were free (an abundant non-economic resource), the producer would not produce beyond that rate at which total product were maximum. He would choose to produce at the boundary between stage II and III.

The only reason for operating in stage I would be insufficient demand to permit taking advantage of increased average productivity. If the fixed factor were divisible into smaller units, the producer could increase production by using fewer units of the fixed factor.

In the above analysis of the three stages of returns in a short-run production situation, no cost information was included. Without information about the relative costs of resources, a producer can do very little in the way of choosing the optimum resource combination, assuming he has some choice and some substitution is possible. However, even without knowledge of costs, provided they are not negative, stage I and III represent underutilization and overutilization, respectively, of the fixed factor, involved.

It would pay to operate in stage II where diminishing returns to factors prevail : this is somewhere between the maximum average product of the variable factor and the maximum total product of the variable factor. Outside stage II, the marginal product of one or the other factor becomes negative.

In stage I, the fixed factor is underutilized and redundant whereas in stage III the fixed factor is overutilized and the variable factor is redundant. Where the marginal product of an output is negative the producer could increase his product by using less of that input.

Symmetry of Regions I and III

As early as 1936, Cassels in his celebrated article (“On the Law of Variable Proportions” in AEA, Readings in the Theory of Income Distribution, Vol. 3) remarked, “The most important thing to observe about this law is that it is symmetrical and consequently the third phase is simply the converse of the first.” Here two questions arise:

(i) What is the practical significance of the symmetrical nature of the law?

(ii) Is it only a theoretical nicety?

Answer to them is that it is not only a theoretical nicety. It has significance in operational aspect. Total output can be increased in either Region I or Regions III by reducing the relative intensity of the excessive factor. In stage I fixed factor is in excess, while in stage III the variable factor is in abundance.

It is important note that there is a confusion while discussing symmetry. Is it factor symmetry or product symmetry? Because of factor symmetry, Regions I and Ill are converse of each other. But this does not mean that the rate of increase of total product in Region I must be symmetrical to the rate of product loss in Region III. Hence there is difference in the exact behaviour and shapes of the total product curve and marginal product curve in Regions I and III. Cassels states,

“Since the proportions of the factors are less affected by each additional unit of the variable factor as we move to the right along the X-axis it is clear that in general the third phase must be more prolonged than the first.”

Long-run Production Function: Returns to Scale

Here we consider the long-run situation where all inputs are variable and can be increased proportionately (according to scale). Returns to scale refer to the way in which output changes when the whole scale of input changes.

The difference between the law of variable proportions and the laws of returns to scale (of plant) can be put as under:

Suppose there are two factors of production K = capital and L = labour. Short-run production function is of the type of:

K+L →P

K+2L → Less than 2P

It means that one unit of K and one unit of L combine to produce one unit of P; while one unit of K and two units of L combine to produce less than two units of P. This is the law of variable proportions. Long-run production function is in the form of:

K+L →P

2K + 2L → 2P or more than 2P or less than 2P.

It means that one unit of K and one unit of L combine to produce one unit of P. Now the units of both K and L are doubled. In that case the output P might be doubled, or may be more than double or may be less than double. This is the case of the laws of returns to the scale of plant, or in short, returns to scale.

Constant Returns to Scale

Stigler says that the first approximation to the returns to the scale of plant is constant returns to scale. If units of all inputs are doubled, the output is also likely to the doubled. Thus

if K+L result in P,

then 2K + 2L result in 2P.

It means that no gain or loss in physical efficiency occurs as the scale of operation is increased. But when physical efficiency increases or decreases as the scale of operation is expanded, we get increasing or decreasing returns to scale.

Increasing Returns to Scale

The decreasing returns to scale cannot be accounted for on the ground that the quality of the input units deteriorates. It is so because it is assumed that the input units are homogeneous. One might reasonably question why increasing or decreasing returns shall occur if the process or operation is expanded according to scale, except as a result of relaxing the assumption of homogeneous units.

Relaxing this assumption would explain, for example, lower productivity (say, yield per acre on farm output) as less fertile acres are added and combine in the same ratio with other inputs of uniform quality. Higher productivity an operation expands according to scale might be explained by the experience or learning of factor that improves the quality and efficiency of certain inputs with time.

Specialization of process and division of labour account for some cases of increasing returns to scale.

There are cases of increasing returns to scale that result from certain physical relationships. Larger diameter of pipe lines may permit increased rates of oil flow per unit of pumping energy because of the small energy loss due to friction. The materials necessary to enlarge the storage capacity of a warehouse are related to the square of the surface dimensions, whereas the actual increase in storage capacity is related to the cube of the surface dimensions.

Care is needed in making pre-mature judgments of increasing returns to scale. Increased physical efficiency of operations in developed economies, sometimes used as an illustration of increasing returns to scale, is often the result of the improved quality of input units, the availability of new complementary inputs (whether internal or external to the firm’s operations), or unavoidable excess capacities in certain inputs at low operating rates.

Let us discuss another cause of increasing returns. The physical characteristic of particular processes requires certain optimal sized components; the relative sizes of tractor motor, for example, to tractor weight to land size. Discrepancies between actual and optimal input sizes do occur since certain inputs are simply not available in certain sizes.

This problem is the indivisibility of input units, as contrasted to adaptability which refers to malleability of form of the inputs rather than to size of the input unit. Indivisibility may be the cause of both increasing and decreasing returns.

The following are the chief causes of increasing returns:

(a) existence of excess capacity in plant;

(b) bulk purchases possible at lower prices, i.e., economies of large-scale buying;

(c) increasing possibility of specialization when scale of production expands; and

(d) indivisibilities.

Decreasing Returns to Scale

The managerial function is often singled out for special treatment as an input, which at some decision-making level such as adaptation to the unexpected, coordination or control, cannot be expanded proportionately. Even if identical units of trained and talented management personnel were available, it is argued that beyond some size, managerial capacity becomes less efficient.

This size would vary for different operations. It changes overtime with changes in technology, such as automation, control systems, advances in management science, computerisation, etc. It is generally believed that decreasing returns to scale arise from the mountains difficulties of coordination and control as scale increases.


There is no one “law” about the behaviour of returns to scale comparable to the Law of Diminishing Returns which eventually comes into play when only one input is varied. It is an empirical issue as to whether they are constant, increasing or decreasing returns to scale.

The phase of increasing returns to scale appears first, but it cannot go on indefinitely. The firm then enters the phase of constant returns to scale for a brief period. Eventually decreasing returns to scale set in.

“What kind of returns is most prevalent in production today?” ask Samuelson and Nordhaus. (Economics, 1995, p. 97.)

The reply to it by them is this: “Economists often think that most production activities should be able to attain constant returns to scale. They reason that if production can be adjusted by simply replicating existing plants over and over again, then the producer would simply be multiplying both inputs and outputs by the same number. In such a case, you would observe constant returns to scale for any level of output.”

Engineering studies reveal that many manufacturing processes enjoy modestly increasing returns to scale for plants up to the largest size used today. But in productive activities involving natural resources, such as growing wine grapes, decreasing returns to scale appear.

All this suggests that, as noted above, there is no one law governing the returns to scale.

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