Production Function

Engineering EconomicsEngineering Economics
(Production Function)(Production Function)

SyllabusSyllabus
• Production Function:Production Function: Meaning, production Meaning, production
with one variable input, the law of variable with one variable input, the law of variable
proportion, the laws of returns to scale. proportion, the laws of returns to scale.
Economies of Scale.Economies of Scale.

Production: MeaningProduction: Meaning
• Any activity which creates value is production.Any activity which creates value is production.
• In other words, production is In other words, production is transformation of inputstransformation of inputs (such (such
as capital, equipment, labour, and land etc ) as capital, equipment, labour, and land etc ) into outputinto output such such
as good or service .as good or service .
• e.g. – transporting sand, operating a jeweller store, drillinge.g. – transporting sand, operating a jeweller store, drilling
for oil, recruiting new employees, designing a system tofor oil, recruiting new employees, designing a system to
measure air pollution, producing biscuits, cultivation,measure air pollution, producing biscuits, cultivation,
trading and so on.trading and so on.

Production-FunctionProduction-Function
• In economics, the technical law, relating inputs to outputs, has been given theIn economics, the technical law, relating inputs to outputs, has been given the
name of production- functionname of production- function..
•    In simple words, production - function expresses the relationship between the In simple words, production - function expresses the relationship between the
physical inputs and  physical output of a firm for a given state of technology.physical inputs and  physical output of a firm for a given state of technology.
• The production-function can be written mathematically as follows:The production-function can be written mathematically as follows:
• qqxx = f (F= f (F11 , F, F22 , F, F33 ……….. F……….. Fnn))
•   HereHere, q, qxx = the quantity of x commodity= the quantity of x commodity
•   FF11 , F, F22 , F, F33 ……….. F……….. Fnn = Different factor-inputs= Different factor-inputs
• This equation tells that the output of x depends on the factors FThis equation tells that the output of x depends on the factors F11 , F, F22 , F, F33
……….. F……….. Fnn ,etc.,etc.
• There is functional relationship between factor-inputs and the amount of goods x.There is functional relationship between factor-inputs and the amount of goods x.
• For example, the output of cloth depends on cotton, thread, machine, labour,For example, the output of cloth depends on cotton, thread, machine, labour,
chemicals, etc.chemicals, etc.
• Hence the relationship between factor-inputs (e.g. thread, machine, labour, Hence the relationship between factor-inputs (e.g. thread, machine, labour,
chemicals, etc.) and the output of cloth can be shown with the help of production-chemicals, etc.) and the output of cloth can be shown with the help of production-
function.function.

Production FunctionProduction Function
• Production function express the technological relationship Production function express the technological relationship
between physical inputs and physical output of a firm under between physical inputs and physical output of a firm under
given technology. given technology.
• A production function may be write as follows A production function may be write as follows
Where,Where,
• Q = output (total product) Q = output (total product)
• Q = f(N, L, K, E,…..)Q = f(N, L, K, E,…..)
• N (land), L(labour), K(capital), E(entrepreneurship) , ..N (land), L(labour), K(capital), E(entrepreneurship) , ..
are the inputs.are the inputs.

Production FunctionProduction Function

Inputs (Factors) of Production/ FactorInputs (Factors) of Production/ Factor
Inputs/Factors/InputsInputs/Factors/Inputs
• Factors of production are broadly classified as :Factors of production are broadly classified as :
• Land:Land:
• Anything which is gift of nature and not the result of human effortAnything which is gift of nature and not the result of human effort, ,
e.g. soil, water, forests, minerals. Owner of land is called landlord. Reward e.g. soil, water, forests, minerals. Owner of land is called landlord. Reward
of land is called as of land is called as rent.rent.
• Labour:Labour:
• Physical or mental effort of human beings that undertakes thePhysical or mental effort of human beings that undertakes the
production process.production process. Labour is supplied by the workers. Labour can be Labour is supplied by the workers. Labour can be
skilled as well as unskilled, physical or intellectual.skilled as well as unskilled, physical or intellectual.
• Reward/price of labour is called as Reward/price of labour is called as wages/ salary.wages/ salary.

Factors Of ProductionFactors Of Production

Inputs (Factors) of Production/ FactorInputs (Factors) of Production/ Factor
Inputs/Factors/InputsInputs/Factors/Inputs
• Capital:Capital:
• Wealth which is used for further production as machine/Wealth which is used for further production as machine/
equipment/intermediary good.equipment/intermediary good. It is outcome of human It is outcome of human
efforts meaning capital is man-made. Reward of capital is efforts meaning capital is man-made. Reward of capital is
called as called as interest,interest,
• Enterprise/Entrepreneurship/organisation:Enterprise/Entrepreneurship/organisation:
• The ability and action to take risk of collecting,The ability and action to take risk of collecting,
coordinating, and utilizing all the factors of production forcoordinating, and utilizing all the factors of production for
the purpose of uncertain economic gains.the purpose of uncertain economic gains. Owner of Owner of
enterprise is entrepreneur. Reward of entrepreneurship is enterprise is entrepreneur. Reward of entrepreneurship is
called as called as profit.profit.

Concept of TimeConcept of Time
• Alfred Marshal introduced the element of time in productionAlfred Marshal introduced the element of time in production
decision. Time can categorize as under:decision. Time can categorize as under:
• Market Period or Very Short Period:Market Period or Very Short Period:
• Market Period or Very Short Period is a period during which allMarket Period or Very Short Period is a period during which all
factors of production and hence cost remains fixed.factors of production and hence cost remains fixed. As such, As such,
outputs as well as supply also remain fixed.outputs as well as supply also remain fixed.
• Short Run:Short Run:
• Short run a period so brief that the amount of at least one input isShort run a period so brief that the amount of at least one input is
fixed.fixed. Thus we have both fixed as well as variable factors.Thus we have both fixed as well as variable factors.
• Long Run:Long Run:
• Long Run is a period of time sufficient enough for all inputs (orLong Run is a period of time sufficient enough for all inputs (or
factors of production), to be variable as far as an individual firm isfactors of production), to be variable as far as an individual firm is
concerned.concerned.
• The length of time necessary for all inputs to be variable may differ The length of time necessary for all inputs to be variable may differ
according to the nature of the industry and the structure of a firmaccording to the nature of the industry and the structure of a firm.

Marshall defined 4 time periods:Marshall defined 4 time periods:
Market periodMarket period - Very short period - Very short period
in which supply is fixed (perfectly in which supply is fixed (perfectly
inelastic). No reflex action of price inelastic). No reflex action of price
on quantity supplied.on quantity supplied.
Short runShort run - A period in which the - A period in which the
firm can change production and firm can change production and
supply but cannot change plant size. supply but cannot change plant size.
Higher prices cause larger quantities Higher prices cause larger quantities
to be supplied.to be supplied.

Long runLong run - Plant size can - Plant size can
vary and all costs become vary and all costs become
variable variable
Secular periodSecular period - (Very - (Very
long run) Permits long run) Permits
technology and population technology and population
to varyto vary

Types of Production-FunctionTypes of Production-Function
Before analyzing the types of production-function it will be useful toBefore analyzing the types of production-function it will be useful to
understand the meaning of following important terms :understand the meaning of following important terms :
A. Fixed Factors and Variable FactorsA. Fixed Factors and Variable Factors
• Factors of production are broadly classified into two categories i.e.Factors of production are broadly classified into two categories i.e.
fixed and variable factors:fixed and variable factors:
• (i) Fixed Factors - The factor inputs which cannot be varied in the(i) Fixed Factors - The factor inputs which cannot be varied in the
short-period, as and when required are called fixed factors.short-period, as and when required are called fixed factors.
• Examples of Fixed Factors are : Plant, machinery, heavy equipments,Examples of Fixed Factors are : Plant, machinery, heavy equipments,
factory building, land etc.factory building, land etc.
• (ii) Variable Factors - The factor inputs which can easily be varied,(ii) Variable Factors - The factor inputs which can easily be varied,
in the short-period as and when required, are called variablein the short-period as and when required, are called variable
factors.factors.
• Examples of variable factors are : labour, raw material, power, fuelExamples of variable factors are : labour, raw material, power, fuel
etc.etc.
• The distinction between fixed factors and variable factors appears onlyThe distinction between fixed factors and variable factors appears only
in the short-period. In the long-run, all the factors of production becomein the short-period. In the long-run, all the factors of production become
variable factors.variable factors.

Fixed Factors and Variable FactorsFixed Factors and Variable Factors

• Short Period And Long PeriodShort Period And Long Period
• The time-period during which a firm in order to makeThe time-period during which a firm in order to make
changes in its production can change only in itschanges in its production can change only in its
variable factors but not in its fixed factors, is termedvariable factors but not in its fixed factors, is termed
as short-period.as short-period. In the short-period, a firm cannotIn the short-period, a firm cannot
change its scale of plant.change its scale of plant.
• The time period in which a firm can change all theThe time period in which a firm can change all the
factors of production and its scale of plant, is termed asfactors of production and its scale of plant, is termed as
long-period.long-period.
• In economics, we study two types of production-In economics, we study two types of production-
functions. In other words, there are two kinds offunctions. In other words, there are two kinds of
input-output relations in production-functions.input-output relations in production-functions.

(i) Short-run Production-functions or the Law of(i) Short-run Production-functions or the Law of
Variable ProportionsVariable Proportions
• In the short period, some factors are fixed and someIn the short period, some factors are fixed and some
of them are variable. What happens when additionalof them are variable. What happens when additional
units of one variable factor of production areunits of one variable factor of production are
combined with a fixed stock of some factors ofcombined with a fixed stock of some factors of
production, is discussed under short-run production-production, is discussed under short-run production-
functions.functions. The law which tells about this relation isThe law which tells about this relation is
called the law of variable proportions or returns to acalled the law of variable proportions or returns to a
factor. Since it is related to a short-period, it is calledfactor. Since it is related to a short-period, it is called
short-run production-function.short-run production-function.

Law of Variable ProportionsLaw of Variable Proportions

ii) Long-run Production-function or Returns to Scaleii) Long-run Production-function or Returns to Scale
• In the long run, all factor-inputs can be varied. ItIn the long run, all factor-inputs can be varied. It
means, that in the long-run, we can expand ormeans, that in the long-run, we can expand or
reduce the scale of production as well. The way inreduce the scale of production as well. The way in
which the output varies with the changes in thewhich the output varies with the changes in the
scale of production is discussed in the long-runscale of production is discussed in the long-run
production-functions.production-functions. The law which states thisThe law which states this
relationship is also called returns to scale.relationship is also called returns to scale. Since it isSince it is
related to the long-period, it is called long-runrelated to the long-period, it is called long-run
production-function.production-function.

Returns to ScaleReturns to Scale

• In this context we have to define three key terms :-In this context we have to define three key terms :-
• (1) Total Product - It refers to the total output of the(1) Total Product - It refers to the total output of the
firm per period of timefirm per period of time
• (2) Average Product - Average Product is total output(2) Average Product - Average Product is total output
per unit of the variable input. Thusper unit of the variable input. Thus Average Product isAverage Product is
total product divided by the number of units of the variabletotal product divided by the number of units of the variable
factor.factor.
• AP = Q/L where Q is Total Product, L is the quantity ofAP = Q/L where Q is Total Product, L is the quantity of
labour.labour.
• (3) Marginal Product - Marginal Product is the change(3) Marginal Product - Marginal Product is the change
in total product resulting from usingin total product resulting from using
• an additional unit of the variable factor.an additional unit of the variable factor.
• MP = dQ/dL, where d is the rate of changeMP = dQ/dL, where d is the rate of change

Concept of ProductConcept of Product
• There are three concepts of product such as:There are three concepts of product such as:
• Total ProductTotal Product
• Average ProductAverage Product
• Marginal ProductMarginal Product

Total ProductTotal Product
• Total Product refers (TP)Total Product refers (TP) to the total volumeto the total volume
of goods and services produced by a firmof goods and services produced by a firm
during a given period of time.during a given period of time.
• Where L is quantity of a factorWhere L is quantity of a factor
• AP is the average productivity of a factor.AP is the average productivity of a factor.
• MP is marginal productivity of a factorMP is marginal productivity of a factor .

Average ProductAverage Product
• Average Product (AP)Average Product (AP) is output (total product) peris output (total product) per
unit of a factor.unit of a factor.
Where,Where,
• i.e. APi.e. AP
• Q is total productQ is total product
• L is quantity of a factor inputL is quantity of a factor input

Marginal ProductMarginal Product
• Marginal Product (MP) is rate of change in total productMarginal Product (MP) is rate of change in total product
with respect to a factor.with respect to a factor.
• Concept of Product contdConcept of Product contd..
• In other words, marginal product is the addition to totalIn other words, marginal product is the addition to total
product by utilizing one more one unit of variable input to theproduct by utilizing one more one unit of variable input to the
production process, keeping other factor fixed.production process, keeping other factor fixed.
Where ,Where ,
• dQ is change in total productdQ is change in total product
• dL is change in quantity of a factor input.dL is change in quantity of a factor input.

Law of Variable Proportions orLaw of Variable Proportions or
Returns to a FactorReturns to a Factor
Meaning and DefinitionMeaning and Definition
• The law of variable proportions has an importantThe law of variable proportions has an important
place in economic theoryplace in economic theory..
• This law exhibits the short-run production-functionsThis law exhibits the short-run production-functions
in which one factor is variable and others are fixed.in which one factor is variable and others are fixed.
• The extra output obtained by applying extra unitThe extra output obtained by applying extra unit of aof a
variable factor can be greater than,variable factor can be greater than, equal to or less thanequal to or less than
the output obtained by its previous unit.the output obtained by its previous unit.
• If the number of units of a variable factor is increased,If the number of units of a variable factor is increased,
the way wherein the output changesthe way wherein the output changes is the concern ofis the concern of
this law.this law.

Law of Variable Proportions orLaw of Variable Proportions or
Returns to a FactorReturns to a Factor
• Thus it refers to the effect of changing factor-ratio onThus it refers to the effect of changing factor-ratio on
the output.the output.
• In short, the law which exhibits the relationship betweenIn short, the law which exhibits the relationship between
the units of a variable factor (keeping all other factors asthe units of a variable factor (keeping all other factors as
constant) and the amount of output in the short-run isconstant) and the amount of output in the short-run is
known as returns to a variable factor.known as returns to a variable factor.
• Thus the law of variable proportions is also named as (orThus the law of variable proportions is also named as (or
returns to a factor) returns to a variable factor.returns to a factor) returns to a variable factor.
• The law states that with the increase in a variable factor,The law states that with the increase in a variable factor,
keeping other factors constant, total product increases at ankeeping other factors constant, total product increases at an
increasing rate, then increases at diminishing rate and finallyincreasing rate, then increases at diminishing rate and finally
starts declining.starts declining.

Reason as to why it is called the Law ofReason as to why it is called the Law of
Variable Proportions:Variable Proportions:
• The factor- proportion (or factor-ratio) varies as oneThe factor- proportion (or factor-ratio) varies as one
input varies and all others are constantinput varies and all others are constant..
• This can be understood with the help of an example.This can be understood with the help of an example.
• Suppose in the beginning 10 acres of land and 1 unitSuppose in the beginning 10 acres of land and 1 unit
of labour are taken for production, hence land-labourof labour are taken for production, hence land-labour
are taken for production, hence land-labour ratio wasare taken for production, hence land-labour ratio was
10: 1.10: 1.
• Now if the land remains the same but the units of labourNow if the land remains the same but the units of labour
increases to 2, now the land-labour ratio would become 5:increases to 2, now the land-labour ratio would become 5:
1.1.

Reason as to why it is called the Law ofReason as to why it is called the Law of
Variable Proportions:Variable Proportions:
• Thus, this law analyses the effects of change inThus, this law analyses the effects of change in
factor-proportions on the amount of output and is,factor-proportions on the amount of output and is,
therefore, called the law of variable proportions.therefore, called the law of variable proportions.

Explanation of the LawExplanation of the Law
• The law of variable proportions can be illustratedThe law of variable proportions can be illustrated
with the help of the following example and diagramwith the help of the following example and diagram

Explanation of the LawExplanation of the Law
• In this example, we assume that land is the fixedIn this example, we assume that land is the fixed
factor and labour is a variable factor.factor and labour is a variable factor.
• The table shows the different amounts of outputThe table shows the different amounts of output
obtained by applying different units of labour toobtained by applying different units of labour to
one acre of land which continues to be fixed.one acre of land which continues to be fixed.

DiagramDiagram
• The law of variable proportions can be explainedThe law of variable proportions can be explained
with the help of diagram below.with the help of diagram below.
• In order to make simple presentation we have drawn aIn order to make simple presentation we have drawn a
TPP (Total Physical Product) curve and a MPPTPP (Total Physical Product) curve and a MPP
(Marginal Physical Product) curve(Marginal Physical Product) curve as smoothas smooth
curves in the diagram, against the variable input,curves in the diagram, against the variable input,
labour.labour.

Three Stages of the LawThree Stages of the Law

Significance of theSignificance of the
Three Stages of the LawThree Stages of the Law
• With the knowledge of the three stages of the lawWith the knowledge of the three stages of the law, a, a
producer can choose the appropriate stage of its operationproducer can choose the appropriate stage of its operation

Reason for Operation of the LawReason for Operation of the Law
• In the short-period all factors of production cannot beIn the short-period all factors of production cannot be
varied.varied.
• Here one is variable factor and others are fixed factors.Here one is variable factor and others are fixed factors.
• There is an optimum combination of different factorsThere is an optimum combination of different factors
that gives the maximum output.that gives the maximum output.
• When there is increase in the units of a variable factorWhen there is increase in the units of a variable factor
before the point of optimum combination, the factorbefore the point of optimum combination, the factor
proportion becomes more suitable and fixed factors areproportion becomes more suitable and fixed factors are
more efficiently utilized.more efficiently utilized.
• Hence it increases the marginal physical product.Hence it increases the marginal physical product.
• In the initial stages the total product may rise at anIn the initial stages the total product may rise at an
increasing rate when we employ more units of a variableincreasing rate when we employ more units of a variable
factor to the fixed factors.factor to the fixed factors.

Reason for Operation of the LawReason for Operation of the Law
• But later, when we employ more units of aBut later, when we employ more units of a
variable factor beyond this optimum combination,variable factor beyond this optimum combination,
the factor proportion becomes unsuitable andthe factor proportion becomes unsuitable and
inefficientinefficient; hence the marginal product of that; hence the marginal product of that
variable factor declines.variable factor declines.
• The quantity of the fixed factor-input per unit ofThe quantity of the fixed factor-input per unit of
the variable input falls as more and more of thethe variable input falls as more and more of the
latter is put to use.latter is put to use.
• Successive units of the variable input, therefore, mustSuccessive units of the variable input, therefore, must
add decreasing amounts to the total output as theyadd decreasing amounts to the total output as they
have less of the fixed input to work with.have less of the fixed input to work with.

• In the long run, all factors are variable.In the long run, all factors are variable.
• The expansion of output may be achieved by varying all factor-inputs.The expansion of output may be achieved by varying all factor-inputs.
• When there are changes in all factor-inputs in the same proportion,When there are changes in all factor-inputs in the same proportion,
the scale of production (or the scale of operation) also gets changed.the scale of production (or the scale of operation) also gets changed.
• Thus, the change in scale means that all factor inputs are changed inThus, the change in scale means that all factor inputs are changed in
the same proportion.the same proportion.
• The term returns to scale refers to the changes in output as all factor-inputsThe term returns to scale refers to the changes in output as all factor-inputs
change in the same proportion in the long run.change in the same proportion in the long run.
• The law expressing the relationship between varying scales of productionThe law expressing the relationship between varying scales of production
(i.e. change of all factor inputs in the same proportion) and quantities of(i.e. change of all factor inputs in the same proportion) and quantities of
output.output.
• The increase in output may be more than, equal to, or less thanThe increase in output may be more than, equal to, or less than
proportional to the increase in factor-inputs.proportional to the increase in factor-inputs.
• Accordingly, returns to scale are also of three types - increasing returns toAccordingly, returns to scale are also of three types - increasing returns to
scale, constant returns to scale and diminishing returns to scale.scale, constant returns to scale and diminishing returns to scale.

• The law of returns to scale with its all the three stages (orThe law of returns to scale with its all the three stages (or
types)types) is shown in the following example and diagram below:is shown in the following example and diagram below:

• The law of returns to scale can also be shown with the helpThe law of returns to scale can also be shown with the help
of a very simple diagram which is given below.of a very simple diagram which is given below. From A to BFrom A to B
in the diagram is the stage of increasing returns; from B to Cin the diagram is the stage of increasing returns; from B to C
constant returns, and from C to D is the diminishing returns toconstant returns, and from C to D is the diminishing returns to
scale.scale.

Causes for the Operation of Returns to ScaleCauses for the Operation of Returns to Scale
• Returns to scale occur mainly because of two reasonsReturns to scale occur mainly because of two reasons

• Thus, the main reason for the operation of the different formsThus, the main reason for the operation of the different forms
of returns to scale is found in economies and diseconomies.of returns to scale is found in economies and diseconomies.
• —— When economies exceed the diseconomies → the stage ofWhen economies exceed the diseconomies → the stage of
increasing returns operate.increasing returns operate.
• —— When economies equal diseconomies → the stage ofWhen economies equal diseconomies → the stage of
constant returns to scale.constant returns to scale.
• —— when diseconomies exceed the economies → stage ofwhen diseconomies exceed the economies → stage of
diminishing returns to scale.diminishing returns to scale.

Distinction between Returns to a Variable FactorDistinction between Returns to a Variable Factor
(or Law of Variable Proportions) and Returns(or Law of Variable Proportions) and Returns
to Scaleto Scale
• The main differences between returns to a variable factor andThe main differences between returns to a variable factor and
returns to scale are as indicated below:returns to scale are as indicated below:

ReferencesReferences
• Engineering Economic Analysis –NPTELEngineering Economic Analysis –NPTEL
http://nptel.ac.in/courses/112107209/http://nptel.ac.in/courses/112107209/
• Engineering EconomicsEngineering Economics
http://www.inzeko.ktu.lt/index.php/EEhttp://www.inzeko.ktu.lt/index.php/EE
• Fundamentals of Economics and ManagementFundamentals of Economics and Management
Institutes of Cost Accountants of IndiaInstitutes of Cost Accountants of India www.icmai.inwww.icmai.in
• Modern Economics : Dr. H. L. AhujaModern Economics : Dr. H. L. Ahuja
• Micro Economics Robert S Pindyck, Daniel L Rubinfeld, PearsonMicro Economics Robert S Pindyck, Daniel L Rubinfeld, Pearson

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