production function give

8/8/2019 Production Function Give

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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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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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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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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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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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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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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



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Average Product


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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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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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

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