Production function [ management ]

Chapter 3: PRODUCTION FUNCTION
Chapter 3: Contents:
 Market Structures
 Price – Output Decision in case of Perfect and Monopoly
 Production Function
 Isoquants and Isocosts
 MRTS Least Cost Combination of Inputs
 Law Of Return , Economies of Scale

1. INTRODUCTION TO PRODUCTION FUNCTION :
A production function shows the relationship between inputs of capital and labour and other
factors and the outputs of goods and services.Production of goods requires resources or
inputs. These inputs are called factors of production named as land, labour, capital and
organization. What Is Production Function in Economics with one or two variables input? A
rational producer is always interested that he should get the maximum output from the set of
resources or inputs available to him. He would like to combine these inputs in a technical
efficient manner so that he obtains maximum desired output of goods.
The production function represents that how much output can be produced by using the
various combinations of the various factors of productions. The theory of production revolves
around the production function.
“A production function can be an equation, table or graph presenting the maximum
amount of a commodity that a firm can produce from a given set of inputs during a period
of time.”
The production function considers the combination of various inputs in the company related
to the production factor considering the four factors of production i.e. land, labour, capital
and entrepreneur. The production function considers the various forms of technology to be
used for the production area which helps to reduce the cost of the production and increases
the output of the product.
The equation of the production function can be expressed in terms of equation is as follows:
Q = ƒ (N,L,K,E,T)
In the equation the factors which are included decides the Q= Quantity supplied with the
factors of production which includes Land, Labour, Capital, and entrepreneur so by which it
can be concluded that the combinations of factors of production the level of output is decided.
In general, economic output is not a (mathematical) function of input, because any given set
of inputs can be used to produce a range of outputs. To satisfy the mathematical definition of
a function, a production function is customarily assumed to specify the maximum output
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obtainable from a given set of inputs. The production function, therefore, describes a
boundary or frontier representing the limit of output obtainable from each feasible
combination of input. (Alternatively, a production function can be defined as the specification
of the minimum input requirements needed to produce designated quantities of output, given
available technology.)
By assuming that the maximum output, which is technologically feasible, from a given set of
inputs, is obtained, economists are abstracting away from technological, engineering and
managerial problems associated with realizing such a technical maximum, to focus
exclusively on the problem of allocative efficiency, associated with the economic choice of
how much of a factor input to use, or the degree to which one factor may be substituted for
another. In the production function itself, the relationship of output to inputs is non-monetary;
that is, a production function relates physical inputs to physical outputs, and prices and costs
are not reflected in the function.
In the decision frame of a firm making economic choices regarding production—how much
of each factor input to use to produce how much output—and facing market prices for output
and inputs, the production function represents the possibilities afforded by an exogenous
technology. Under certain assumptions, the production function can be used to derive a
marginal product for each factor. The profit-maximizing firm in perfect competition (taking
output and input prices as given) will choose to add input right up to the point where the
marginal cost of additional input matches the marginal product in additional output. This
implies an ideal division of the income generated from output into an income due to each
input factor of production, equal to the marginal product of each input.
The inputs to the production function are commonly termed factors of production and may
represent primary factors, which are stocks. Classically, the primary factors of production
were Land, Labour and Capital. Primary factors do not become part of the output product, nor
are the primary factors, themselves, transformed in the production process. The production
function, as a theoretical construct, may be abstracting away from the secondary factors and
intermediate products consumed in a production process. The production function is not a full
model of the production process: it deliberately abstracts from inherent aspects of physical
production processes that some would argue are essential, including error, entropy or waste,
and the consumption of energy or the co-production of pollution. Moreover, production
functions do not ordinarily model the business processes, either, ignoring the role of strategic
and operational business management. (For a primer on the fundamental elements of
microeconomic production theory, see production theory basics).
The production function is central to the marginalist focus of neoclassical economics, its
definition of efficiency as allocative efficiency, its analysis of how market prices can govern
the achievement of allocative efficiency in a decentralized economy, and an analysis of the
distribution of income, which attributes factor income to the marginal product of factor input.
The firm is assumed to be making allocative choices concerning how much of each input
factor to use and how much output to produce, given the cost (purchase price) of each factor,
the selling price of the output, and the technological determinants represented by the
production function.

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2. LONG RUN – SHORT RUN PROCESS:

The production function decides the process of the production and according to the process
cycle. The production function helps the manager to take various decisions related to the
business related to the production so that the equilibrium position of the company can be
maintained. Thus the production technique is divided into two parts :
1. LONG RUN FUNCTION
2. SHORT RUN FUNCTION
Thus the process of the function decides the profitability of the company. According to the
function the factors of the production are been decided and according to that the production
cycle of the firms product is decided.
In the short run function the plant size is fixed and the variable factor labour keeps on
changing. Thus the factor is remains fixed then the other factors if kept on increasing it
results into negative return for the firms. Thus in the short run as the firm earns a maximum
production level and after that if the units of production are kept on increasing then the
production function starts giving the negative return as the plant size is fixed so, the labours
becomes more and so the machines can be used more or the labours remains ideal in the firm.
Thus the company earns expenses on the production cycle but the production is not earned as
per the expectation which results in loss area in the production function resulting into
negative returns. Thus in the short run function the firms focuses on production function up to
a certain level of profitability return.
In a long run production function the plant size as well as the labour size as variable. Thus the
company’s analysis the production function cycle in the market for the long term and
according the market situation the production of the company is decided. In long term if it is
considered that the land is a variable factor then that means that the plant size is variable in
the planning process and the increment in the land sector results in more productivity which
results in increment in profit. Thus in the long run the planning of the production function
which results in to the profit area according to the situation.
Thus both the long run and the short run production function is to be considered which
ultimately decides the market structure. Thus the production function considers the various
types of the cost which includes average revenue, marginal revenue and the total
productivity.

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There are three major ways to measure the productivity of labour which helps to measure the
productivity of the labour along with which the production function moves in the market.
1. TOTAL PRODUCTIVITY:
Total productivity (also known as total physical product) is defined as the total quantity of
output produced by a firm for a given quantity of input necessities. Total product identifies
the specific outputs which are possible using variable levels of input. An understanding of
total product is essential to the short-run analysis of a firm's production. Changes in total
product are taken into account closely when there are changes in variable costs (labour) of
production.
Thus the total productivity shows the relationship between number of workers and the total
number of outputs been produced which is termed as Q, holding the concept that the other
factors remains constant.
 Thus for e.g. For a coffee shop, output would be measured in ―number of
coffee cups a day‖
 For a steel mill, output would be measured in ―tons of steel produced a day‖
Thus the total productivity shows the three following conditions:

The first situation shows that as the labour increases the number of output also increases in an
increasing, constant or in a decreasing form. But as the three cases are seen the total output
increases thus the total productivity is profitable for the firm. But as the marginal revenue
starts decreasing it give low total productivity which results in losses.
2. Marginal Productivity :
Productivity is, at its most basic, the output gained from a unit of input. For example, a
clothing company’s productivity could be the number of jeans sewn per worker or per hour.
To increase productivity, you have to increase the input. To use the clothing factory example,
you would have to buy more machines, hire more workers or find some means of increasing
efficiency to increase productivity.
The term ―marginal productivity‖ refers to the extra output gained by adding one unit of
labor; all other inputs are held constant. So, the technology and efficiency of the factory stays
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the same. Marginal productivity is the extra jeans sewn, that is output gained, by hiring an
extra worker, for example. The additional output that can be produced by adding one more
worker while holding plant size constant.
MP = ΔQ/ΔL8
Is the slope of the Total Product Function.
3. AVERAGE PRODUCTIVITY:
Average productivity is measured by taking the total output and dividing the quantity by the
number of workers. For example, if the combined number of phone calls handled in a week is
1,300 and the company has 10 employees each working the same shift length, the average
productivity per worker is 130. Businesses use average productivity figures to gain a
perspective on the performance of its workforce: by pooling the labour of every individual, it
focuses less on how to improve a problematic worker's output and more as an estimate of the
output currently being given. If low productivity is the result of a systemic issue within the
company and for reasons that affect all workers, measuring average productivity as opposed
to per-worker productivity is the better plan.
If average productivity is more than marginal productivity, average productivity will
decrease. If the average productivity is less than marginal productivity, average productivity
will increase.
Thus the average productivity shows the average change in the production theory of the
organisation. Thus if the average production decreases and it affects the marginal
productivity which in turn affects the total productivity which frames the law of return.

LAW OF DIMINISHING RETURN :
Diminishing Returns occurs in the short run when one factor is fixed (e.g. Capital) If the
variable factor of production is increased, there comes a point where it will become less
productive and therefore there will eventually be a decreasing marginal and then average
product This is because if capital is fixed extra workers will eventually get in each other’s
way as they attempt to increase production. E.g. think about the effectiveness of extra
workers in a small café. If more workers are employed production could increase but more
and more slowly.
This law only applies in the short run because in the long run all factors are variable. Assume
the wage rate is £10, then an extra worker Costs £10. The Marginal Cost (MC) of a sandwich
will be the Cost of the worker divided by the number of extra sandwiches that are produced
Therefore as MP increases MC declines and vice versa
A good example of Diminishing Returns includes the use of chemical fertilizers- a small
quantity leads to a big increase in output. However, increasing its use further may lead to
declining Marginal Product (MP) as the efficacy of the chemical declines.
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Thus the law of diminishing return states that ―As more of a variable input (labor) is added
to a fixed input (plant), additions to output eventually slow down.” Thus the productivity
is acceptable only upto the level where the total productivity and marginal productivity
increases.
The following example shows the marginal rate of return:
EXAMPLE :

As the law of diminishing considers the labour and productivity to change and other factors
keeps on changing, here we consider the land factor to be fixed in the market. In the example
we can see that the total productivity keeps on changing but then also when the returns are
considered it starts decreasing the return and still the production is continued of the firm it
may result into negative return.
As the total productivity starts from 10 to 68 units the average and marginal units are been
calculated as under:
Average productivity = total productivity/ Labour units, thus 10/1=10 and so on.
Marginal productivity = 22-10= 12 marginal difference of two total productivity.

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Thus as the marginal productivity is calculate the company when inputs 1 or 2 labour it earn
increasing return, for 3 and 4 number of units the return becomes constant and finally for 5
and 6 unit the return has started diminishing. It proves that now suppose if the labour starts
increasing then the firm will result in negative profits.
Thus the law of return says that upto a certain level the firm are increasing and constant
return the firm falls in part of diminishing return and still of the production is continued it
may result in negative return which indicates the loss of the firm.
MRTS :
Prof. R.G.D. Allen and J.R. Hicks introduced the concept of MRS (marginal rate of
substitution) in the theory of demand. The similar concept is used in the explanation of
producers’ equilibrium and is named as marginal rate of technical substitution (MRTS).
This theory finds the combinations of inputs in such a way that the total productivity is been
maintained at a lower production cost in the organisation. This means that input factors are
combined in such a way that the maximum production is achieved at the minimum cost.
This is called least cost combination.
The analysis of production function has shown that alternative combinations of factors of
production, which are technically efficient, can be used to produce a given level of output. Of
these, the firm will have to choose that combination of factors which will cost it the least. In
this way the firm can maximise its profits. The choice of any particular method from a set of
technically efficient methods is an economic one and it is based on the prices of factors of
production at a particular time.
The firm can maximise its profits either by maximising the level of output for a given cost or
by minimising the cost of producing a given output. In either case, the factors will have to be
employed in optimal combination at which the cost of production will be minimum.

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Thus in this method the variables are been such arranged that the company combines the
factors of production and gets the maximum level of output at the given budgeted rate.
In this every input should be combined in such a way that marginal productivity of a factor
and the marginal utility of the money spent on it are equal. Thus in MRTS the cost of
production is minimum and is also known as the optimal combination of input for the
production theory. The equation of MRTS is as follows:

Where MP is the marginal productivity and P stands for price and is continued till the number
of inputs been used by the firm. So a,b,c,d ,n are the factors of inputs of the variables.
Assumption of the MRTS Theory :
1. MRTS works in a prefect competitive market as this theory works in lowering the prices at
the maximum output which is the need of the perfect competitive world rather then the
monopoly companies.
2. This theory considers that the factors of production is going to be mobile thus any factors
can be changed as there is a change in the market situation.
3. The prices of each input variable is been decided and once decided it should not be
changed and all the inputs should consider the similar value as a part of the price.
4. The production theory as works on marginal theory, the marginal rate of individual input
unit is to be considered then only the optimal value is created.
Example:
Suppose a firm, uses A,B,C as inputs in the process of the production. The prices of A ,B and
C are 6,4 and 2 respectively. On all these three Rs.122 has been spent. The Marginal
productivity is as follow:

So here the prices of A is 6 RS , for B is 4 RS and for the c the price is 2 respectively. So to
find the optimal value first the MPA, MPB and MPC is to divided by the prices individually
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so that the optimal combination values are obtained. Thus where the MPA/PA= MPB/PB =
MPC/PC is there that values are been selected.
Thus the calculation of MRTS is as follows:

Thus here for Product A optimum value of combination 3 is achieved at 9 level of units,
whereas for product B the optimum value of combination 3 is achieved at 11 units and for the
product C 12 units of production is required. Thus the optimal combination defines the
maximum production and the minimum price.
Thus the , Minimum Price : A= 9*6 =

54

B= 11*4 =

44

C= 12*2 =

24

Thus the optimal Production is as follows:
A= 30+28+24+18 = 100
B= 24+22+18+12 = 76
C= 20+18+14+6 = 58
Thus according to the MRTS technique the minimum price for the production is 122 at which
the optimal production available for the company is 234 units. Thus the MRTS works for
calculating the expected units of production.
Economies of scale:
he cost advantage that arises with increased output of a product. Economies of scale arise
because of the inverse relationship between the quantity produced and per-unit fixed costs;
i.e. the greater the quantity of a good produced, the lower the per-unit fixed cost because
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these costs are shared over a larger number of goods. Economies of scale may also reduce
variable costs per unit because of operational efficiencies and synergies. Economies of scale
can be classified into two main types: Internal – arising from within the company; and
External – arising from extraneous factors such as industry size.
Economies of scale can arise in several areas within a large enterprise. While the benefits of
this concept in areas such as production and purchasing are obvious, economies of scale can
also impact areas like finance. For example, the largest companies often have a lower cost of
capital than small firms because they can borrow at lower interest rates. As a result,
economies of scale are often cited as a major rationale when two companies announce a
merger or takeover.
However, there is a finite upper limit to how large an organization can grow to achieve
economies of scale. After reaching a certain size, it becomes increasingly expensive to
manage a gigantic organization for a number of reasons, including its complexity,
bureaucratic nature and operating inefficiencies. This undesirable phenomenon is referred to
as "diseconomies of scale".
Internal economies of Scale:
The internal economies of scales are the production advantages which the companies earns
from the internal working of the organisation. Internal economies of scale are the advantages
of large scale production. They are enjoyed by the firm when it increases its scale of
production. They accrue to the firm from their own actions. They affect the shape of the longrun average cost curve. They are responsible for increasing returns to scale. According to
many economists, internal economies arise due to indivisibility of some factors. As the output
increases the large indivisible factors can be used more efficiently and, therefore, the firm
experiences increasing returns to scale.Thus the internal economies of scale includes the
following points:
1. Technical Economies:
The important technical economies result from the use of specialised capital equipment,
which comes into effect only when the output is produced on a large scale. Technical
economies also arise from the indivisibilities, which are the characteristics of the modern
techniques of production. In other words, as the scale of production increases the firm reaps
the advantages of mechanisation of using mass production methods. This will reduce the unit
cost of production.
2. Managerial Economies :
Large scale production makes possible the division of managerial functions. Thus, there
exists a production manager, a sales manager, a finance manager, a personnel manager and so
on in a large firm. However, all or most of the managerial decisions are taken by a single
manager in a small firm. This division of managerial functions increases their efficiency. The
decentralisation of managerial decision making also increases the efficiency of management.
Large firms are also in a position to introduce mechanisation of managerial functions through
the use of telex machines, computers and so on. Hence, as output increases the managerial
costs per unit of output continue to decline.

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3. Marketing Economies :
They are allied with the selling of the product of the firm. They arise from advertising
economies. Since, advertising expenses increase less thanproportionately with the increase in
output, the advertising costs per unit of output fallsas the output increases. Similarly, other
sales promotion expenditures like samples, salesmen force etc. also increase less than
proportionately with the output. Further, a large firm can have special arrangements with
exclusive dealers to maintain a good service department for the product of the firm. Hence,
the average selling costs fall with the increase in the size of the firm.
4. Financial Economies :
The role of finance is to aid the firm in meeting random changes in the input and the output
sides of the operations of the firm. The purpose of inventories is to smooth out the supply of
inputs and the supply of outputs. Inventories on spare parts, raw materials and finished
products increase with the scale of production, but they do not increase proportionately with
the increase in the size of output. Therefore, as the size of output amplifies the firm can hold
smaller percentage of inventories to meet random changes.
External Economies of Scale:
The external economies arise outside the firm as a result of improvement in the industrial
environment in which the firm operates. They are external to the firm, but internal to the
industry to which the firms belong. They may be realised from the actions of other firms in
the same industry or in another industry. Their effect is to cause a change in the prices of
factors employed by the firm. They cause a shift in the short-run and long-run cost curves of
the firm.
The important external economies are the following:
1. Concentration Economies:
Expansion of an industry increases the demand for various kinds of materials and capital
equipment’s. This will lead to large scale production of materials and equipment’s. Large
scale production will reduce their cost of production and therefore, their prices. Hence, the
firms using them will get them at lower prices.Expansion of an industry may lead to the
discovery of new technical know-how. As a result of this the firms may be able to use
improved and better machinery which will increase the productivity of the firms and
therefore, reduce the cost of production. Thus the place where the productional units are been
established and their nearby areas gives the firm the benefit of concentration of economies.
2. Information Economies :
External economies also arise from the interchange of technical information between firms.
With the expansion of an industry the firms may give the information about the technical
knowledge through the publication of trade and technical journals. The firms may also set up
jointly research institutes to develop new improved techniques. Thus the information which is
received is very helpful for the companies development. Thus this advantage is been got by
the large scale companies as compared to the small scale units. For eg. Ahmedabad Textile

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Industry Research Association (ATIRA), is a unit working for the collection of the vital
information which is helpful for the overall development of the firms.
3. Disintegration Economies:
As the industry grows the training facilities for labourwill increase. This helps the
development of skilled labour, which will increase the productivity of workers in the
firms.Expansion of an industry may facilitate the growth of subsidiary and ancillary
industries to produce tools, equipment’s, machines etc. and to provide them to the main
industry at the lower prices. Likewise, firms may also come up to transform the waste of the
industry into some useful products. This tends to reduce the cost of production.The expansion
of an industry may expedite the development of transportation and marketing facilities which
will reduce the cost of transportation. Thus the disintegration facilities help for the overall
advantage for the production areas of the firm.
ISOQUANTS AND ISOCOSTS:
An isoquant shows all those combinations of factors which produce the same level of output.
An isoquant is also known as equal product curve or iso-product curve. Thus the isoquant
curve shows the combinations of the two factors of production for which the total outlay of
the product remains the same. Thus the cost earned for the prodcutional area is called iscosts.
Certain Assumptions of this theory includes :
 Uses capital and labour combination for production
 All other factors remains fixed.
 Production method is given, and no change can be made.
In economics, an isocost line represents all combinations of inputs which cost the same total
amount. Although, similar to the budget constraint in consumer theory, the use of the isocost
line pertains to cost-minimisation in production, as opposed to utilitymaximisation.
For the two production inputs, labour and capital, with fixed unit costs of the inputs, the
equation of the Isocosts line is

Where w represents the wage rate of labour, r represents the interest rate of capital, K is the
amount or units of capital used, L is the amount of labour used and C is the total costof
acquiring these inputs.
PROPERTIES OF ISOQUANTS: (CHSRECTERISTICS):
1. Isoquants slope downwards to the right: It means that, in order to keep the output constant;
when the amount of one factor is increased the quantity of other factor mustbe reduced.

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An upward sloping isoquant demonstrates that a given product can be produced with less of
both the factors of production. An entrepreneur, who is maximising profits, would not use
any combinations of factors shown on an upward sloping portion of an isoquant.
Therefore, the points on the upward sloping portion of an isoquant cannot represent an
equilibrium position. Similarly, a horizontal or vertical range of an isoquant cannot also
represent a possible position of equilibrium. In this case, the same output could be obtained at
a reduced cost by reducing the amount of one of the factors. Thus, isoquants slope
downwards to the right as in fig 3.15.

2. Isoquants are convex to the origin:
The slope, at any point of an isoquant, is negative. Its numerical value measures the marginal
rate of technical substitution between labour and capital. It equals the ratio of the marginal
product of labour to the marginal product of capital. Thus, the slope of an isoquant is

Where ΔK is the change in capital, ΔL is the change in labour, MRTSLK is the marginal rate
of technical substitution of labour for capital, MPL is the marginal product of labour and
MPK is the marginal product of capital.
The convexity of isoquant means that as we move down the curve less and less of capital is
given up for an additional unit of labour so as to keep constant the level of output. This can
be observed from the Fig. 3.16.

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It can be seen from the figure above that as we increase labour at a constant rate the amount
of capital given up (ΔK) for an additional unit of labour goes on falling. Thus, the convexity
of the isoquant shows that the marginal rate of technical substitution of labour for capital is
diminishing.
3. Isoquants do not intersect:
By definition isoquants, like indifference curves, can never cut each other. If they cut each
other it would be a logical contradiction.
4. The higher the isoquant curve the more output it indicates.
5. All points on isoquant curve shows equal production.
ISOQUANT GRAPH:
The firm uses an input as a substitute of another input in such a manner that at each
combinations of inputs output remains same. The list of equal output at different
combinations is prepared so that its forms a convex shapes. Those convex curves are
considered as isoquant curves and the cost related to that combination for production is called
the isocost.

Thus here the various combinations of the labour and capital is considered and based on that
the marginal rate ration is been calculated and it can be observed that all the possible
combinations the variable output remains the same.

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Thus in the above graph:
-

On OX axis the capital units are considered and on OY axis the units of labour is
mentioned.
R,P,N and M shows the various combinations of the isoquant curves which is a
combination of capital and labour.
Thus the convex shapes been formed are considered as isoquant curves which shows
equal production units i.e. 100 at each combines combination.

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