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ProductionFunction
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What is production?
Production is the process that transformsinputs into output.
Production is the process by which theresources (input) are transformed into adifferent and more useful commodity.
Various inputs are combined in differentquantities to produce various levels of output.
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Some Basic Concepts
Production:
Production means transforming inputs ( Labour,Machines, Raw materials etc.) into an output.
Input and Output:
An input is a good or service that goes into theprocess of production. Land, Labour, Capital,Management, Entrepreneur and Technology are
classified as inputs. An output is any good or service that comes out of
the production process.
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FixedInputs & Variable Inputs:
Fixed inputs remains fixed (constant) up to certainlevel of output.
Variable inputs change with the change in output.
Short Run and Long Run:
Short run refers to a period of time in which supply ofcertain inputs i.e., plant, building and machinery etc.is fixed or inelastic.
Long run refers to a time period in which the supplyof all the inputs is elastic or variable.
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Production Function
Production function is defined as the functionalrelationship between physical inputs ( i.e., factors of
production ) and physical outputs, i.e., the quantityof goods produced.
Production function may be expressed as under:
Q= f ( K,L)
Where ;
Q= Output of commodity perunit of time.
K= Capital.
L = Labour.
f= Functional Relationship.
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Production function depends on :
Quantities of recourses used.
State of technical knowledge.
Possible process.
Size of firms.
Relative prices of factors of production.
Combination of factors.
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Production decisions of a firm are similar
to consumer decisions
y Can also be broken down into three steps
Production Technology
Cost Constraints
Input Choices
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Production Decisions of a Firm
1. Production Technology
Describe how inputs can be transformed
into outputs Inputs: land, labor, capital and raw
materials
Outputs: cars, desks, books, etc.
Firms can produce different amounts ofoutputs using different combinations of
inputs
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Production Decisions of a Firm
2. Cost Constraints
Firms must considerprices of labor,
capital and other inputs
Firms want to minimize total productioncosts partly determined by input prices
As consumers must consider budget
constraints, firms must be concerned
about costs of production
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Production Decisions of a Firm
3. Input Choices
y Given input prices and production
technology, the firm must choose howmuch of each inputto use in producing
output
y Given prices of different inputs, the firm
may choose different combinations ofinputs to minimize costs
If labor is cheap, firm may choose to
produce with more labor and less capital
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Managerial uses of production
function
-Least-Cost-Factors combination-Optimum level of output-Programming technique in productionplanning-Equilibrium level of output-Returns to scale
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Short run: Short run refers to a period of time inwhich supply of certain factor inputs is fixed orinelastic.
Long run: Long run refers to a period of time inwhich the supply of all the inputs is elastic, but not
enough to permit a change in technology.
Very long period: Very long period refers to aperiod of time in which along with all other factor
inputs,the technology of production can also bechanged.
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Short run analysis of production function
Laws of Production
Laws of production are of two types:
The law of variable proportions.
Laws of returns to scale.
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Short Run Production Function: The Law of
Variable Proportions
Statement of the law:
The law of variable proportions states that when more
and more units of the variable factor are added to a
given quantity of fixed factors, the total product may
initially increase at an increasing rate reach themaximum and then decline.
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Assumptions
1. The law applies only in the short run.
2. One factor of production is variable & others
are fixed.
3. All units of variable factor are homogeneous.
4. State of technology is given & remains thesame.
5. Factor proportions can he changed.
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Key terms in production analysis
Total product (TP): The total amount of output
resulting from a given production function
Average product(AP): Total product per unit of
given input factor.
Marginal product(MP): The change in total
product per unit change in given input factor.
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Total product / Total physical product :- It
is defined as the total quantity & services
produced by a firm with the given inputs
during a specified period of time or total
product is sum total of output of each unit of
variable factor used in the process ofproduction. Thus
TP =S
um of MPsTP =AP X n
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Marginal Product :- is a net addition to total product when
one more unit of variable factors employed
MP = TPn- TPn-1MP = TP/ L
Average. product :- is the per unit production of the
variable factors i.e. AP = TP/ L
Relationship between TP & MP
1. When TP increases at increasing rate, MP also increases.
2. When TP starts increasing at decreasing rate, MPdecreases but remains positive
3. When TP is maximum & constant MP is O (zero)
4. When TP begins to fall, MP is negative
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Marginal and Average Product
When marginal product is greater than the
average product, the average product is
increasing
When marginal product is less than theaverage product, the average product is
decreasing
When marginal product is zero, total
product (output) is at its maximum
Marginal product crosses average product
at its maximum
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Relationship between AP & MP
1. Both AP & MP cures are derived from TP since, AP =TP/ L & MP = TP/L
2. When MP is greater than AP, AP rises but MP rises at
faster pace.
3. When MP equals to AP, AP is constant
4. When MP is less than AP, AP falls but MP falls at higher
rate.
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Three stages of production
-Stage I: Increasing Returns TP increases atincreasing rate, indicated by increasing MP.
-There is intermediary constant stage between
stage I & stage II. TP increases at a constant rate
indicated by constant MP
-Stage II:Diminishing Returns TP continues to
increase but at diminishing rates, indicated by
declining MP
-Stage III: Negative Returns TP begins to decline,
indicated by negative MP
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Three stages of production
Total Product Marginal Product Average Product
STAGE I
Increases at an increasing
rate
Increases and reaches its
maximum
Increases (but slower than
MP)
STAGE II
Increases at a diminishing
rate and becomes
maximum
Starts diminishing and
becomes equal to zero
Starts diminishing
STAGE III
Reaches its maximum,
becomes constant and then
starts declining
Keeps on declining and
becomes negative
Continues to diminish (but
must always be greater
than zero)
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Three Stages of Production in Short Run
AP,MP
X
Stage IStage II
Stage III
APX
MPXFixed input grosslyunderutilized;
specialization andteamwork causeAP to increasewhen additional Xis used
Specialization andteamwork continue to
result in greateroutput whenadditional X is used;fixed input beingproperly utilized
Fixed input capacityis reached;additional X causesoutput to fall
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Factors behind the law
-- Stage I & II ( up to optimum fixed & variable
factor combination )
- Indivisibility of fixed factors- Division of labour
-- Stage III
- Improper substitution of variable factor for fixed
factor
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1. Increasing return to a factor:-
(i) Fuller utilization of fixed factor : In the initialstages Fixed factor remain under utilized its fuller
utilization starts with the more application of
variable factor, hence, initially additional unit of
variable factors add more to the total output
(ii) Specialization ofLabour :- Additional
application ofVariable factor causes process based
division of Labour that raises the efficiency of factors.Accordingly marginal productivity tends to rise.
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2. Diminishing return to a factor:-
(i) Imperfect factor substitutability :- Factors ofproduction are imperfect substitutes of each other.
More & more of Labour, for eg. Cannot be continuously
used in place of additional capital.Accordingly
diminishing returns to variable factor becomes
inevitable.
(ii) Disturbing the optimum proportion :-Continuous increase in application of variable factor
along with fixed factors beyond a point crosses the limit
of ideal factor ratio. This results
in poor co-ordination between the fixed & variable
factors which causes diminishing return to a factor.
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3. Negative returns to a factor :-
(i) Overcrowding :- When more & more variablefactors are added to a given quantity of fixed
factor it will lead to over crowding & due to this
MP of the Labours decreases & it goes into
negative
(ii) Management Problems :- When there are too
many workers they may shift the responsibility to
others & it becomes difficult for the management tocoordinate with them. The Labours avoid doing
work. All these things lead to decrease in efficiency
of Laboures. Thus the output also decreases.
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Production: One Variable Input
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Production: One Variable Input
Observations:
1. When labor is zero, output is zero as well
2. With additional workers, output (q)
increases up to 8 units of labor
3. Beyond this point, output declines
Increasing labor can make better use of
existing capital initially After a point, more labor is not useful and
can be counterproductive
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Production: One Variable Input
Average product of Labor- Output per
unit of a particular product
Measures the productivity of a firmslabor in terms of how much, on
average, each worker can produce
L
q!!
InputLabor
OutputAPL
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Production: One Variable Input
Marginal Product of Labor additional
output produced when labor increases
by one unit
Change in output divided by the change
in labor
Lq
(
(!(
(!InputLabor
OutputMPL
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Production: One Variable Input
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Production: One Variable Input
We can graph the information in Table to
show
y How output varies with changes in labor
Output is maximized at 112 units
y Average and Marginal Products
Marginal Product is positive as long as
total output is increasing Marginal Product crosses Average Product
at its maximum
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At point D, output is
maximized.
Labor per Month
Output
per
Month
0 2 3 4 5 6 7 8 9 101
Total Product
60
112
A
B
C
D
Production: One Variable Input
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Average Product
Production: One Variable Input
10
20
Output
per
Worker
30
80 2 3 4 5 6 7 9 101 Labor per Month
E
Marginal Product
Left of E: MP >AP &AP is increasing
Right of E: MP
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Law of DiminishingMarginal Returns
When the use of labor input is small and
capital is fixed, output increases considerably
since workers can begin to specialize and MPof labor increases
When the use of labor input is large, some
workers become less efficient and MP of labor
decreases
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Law of DiminishingMarginal
Returns Typically applies only for the short run
when one variable input is fixed
Can be used for long-run decisions toevaluate the trade-offs of different plant
configurations
Assumes the quality of the variable
input is constant
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Law of DiminishingMarginal
Returns
Easily confused with negative returns
decreases in output Explains a decliningmarginal product,
not necessarily a negative one
y Additionaloutput can be declining while
totaloutput is increasing
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Law of DiminishingMarginal
Returns
Assumes a constant technology
y
Changes in technology will cause shifts inthe total product curve
y More output can be produced with same
inputs
y Labor productivity can increase if there areimprovements in technology, even though
any given production process exhibits
diminishing returns to labor
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The Effect ofTechnological
ImprovementOutput
50
100
Labor per
time period0 2 3 4 5 6 7 8 9 101
A
O1
C
O3
O2
B
Moving from A to B to C, labor
productivity is increasing over time
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Production: Two Variable
Inputs
Firm can produce output by combining
different amounts of labor and capital
In the long run, capital and labor are
both variable
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Production: Two Variable Inputs
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The information can be represented
graphically using isoquants
y Curves showing all possible combinations of
inputs that yield the same output
Curves are smooth to allow for use offractional inputs
y Curve 1 shows all possible combinations of
labor and capital that will produce 55 units of
output
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Isoquant Map
Labor per year1 2 3 4 5
Ex: 55 units of output
can be produced with
3K & 1L (pt. A)
OR
1K & 3L (pt. D)
q1= 55
q2= 75q3= 90
1
2
3
4
5Capital
per year
D
E
A B C
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Production: Two Variable Inputs
Diminishing Returns to Labor with
Isoquants
Holding capital at 3 and increasinglabor from 0 to 1 to 2 to 3
y Output increases at a decreasing rate (0,
55, 20, 15) illustrating diminishing marginal
returns from labor in the short run and longrun
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Diminishing Returns to Capital with
Isoquants Holding labor constant at 3 increasing
capital from 0 to 1 to 2 to 3
y Output increases at a decreasing rate (0,
55, 20, 15) due to diminishing returns from
capital in short run and long run
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Diminishing Returns
Labor per year1 2 3 4 5
Increasing labor holding
capital constant (A, B,
C)
OR
Increasing capitalholding labor constant
(E, D, C
q1= 55
q2= 75q3= 90
1
2
3
4
5Capital
per year
D
E
A B C
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SubstitutingAmong Inputs
y Companies must decide what combination
of inputs to use to produce a certain
quantity of output
y There is a trade-off between inputs,
allowing them to use more of one input and
less of another for the same level of output
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SubstitutingAmong Inputs
y Slope of the isoquant shows how one input
can be substituted for the other and keep
the level of output the same
y The negative of the slope is the marginal
rate of technical substitution (MRTS)
Amount by which the quantity of one inputcan be reduced when one extra unit of
another input is used, so that output remains
constant
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The marginal rate of technicalsubstitution equals:
)( qL
KMRTS
InputLaborinChange
InputCapitalinChangeMRTS
oflevelfixedafor(
(!
!
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As labor increases to replace capital
y Labor becomes relatively less productive
y Capital becomes relatively more productive
y Need less capital to keep output constant
y Isoquant becomes flatter
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Marginal Rate ofTechnical Substitution
Labor per month
1
2
3
4
1 2 3 4 5
5Capitalper year
Negative Slope measures MRTS;
MRTS decreases as move down
the indifference curve
1
1
1
1
2
1
2/3
1/3
1=55
Q2=75
Q3=90
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Isoquants: Special Cases
Two extreme cases show the possible
range of input substitution in
production1. Perfect substitutes
y MRTS is constant at all points on isoquant
y Same output can be produced with a lot
of capital or a lot of labor or a balancedmix
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Perfect Substitutes
Labor
per month
Capital
per
month
Q1 Q2 Q3
A
B
C
Same output can be
reached with mostly
capital or mostly labor (A
or C) or with equal
amount of both (B)
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2. Perfect Complements
y Fixed proportions production function
y There is no substitution available between
inputs
y The output can be made with only a specific
proportion of capital and labor
y Cannot increase output unless increase both
capital and labor in that specific proportion
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Fixed-Proportions Production Function
Labor
per month
Capitalper
month
K1 Q1A
Q2
Q3
B
C
Same output can
only be produced
with one set ofinputs.
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Production Function in Long Run
Laws of Returns to Scale
Thepercentage increase in output when all inputs vary
in the sameproportion is known as returns to scale. It
obviously relates to gr eater use of inputs maintaining
the same technique of production.
Three Situations of Returns To Scale
- Increasing Returns to Scale
- Constant Returns to Scale
- Decreasing Returns to Scale
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Returns to Scale
Increasing returns to scale: output
more than doubles when all inputs are
doubledy Larger output associated with lower cost
(cars)
y One firm is more efficient than many
(utilities)y The isoquants get closer together
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Increasing Returns to Scale
10
20
30
The isoquants
move closer
together
Labor (hours)5 10
Capital
(machine
hours)
2
4
A
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Returns to Scale
Constant returns to scale: output
doubles when all inputs are doubled
y Size does not affect productivityy May have a large number of producers
y Isoquants are equidistant apart
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Returns to Scale
Constant Returns:
Isoquants are
equally spaced2
0
30
Labor (hours)155 10
A
10
Capital
(machine
hours)
2
4
6
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Returns to Scale
Decreasing returns to scale: output
less than doubles when all inputs are
doubledy Decreasing efficiency with large size
y Reduction of entrepreneurial abilities
y Isoquants become farther apart
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Returns to Scale
Labor (hours)
Capital
(machine
hours)
Decreasing Returns:Isoquants get further
apart
1020
10
4
A
30
5
2