production function

PRODUCTION ANALYSISPRODUCTION ANALYSIS

ProductionProductionAn entrepreneur must put together

resources -- land, labour, capital -- and produce a product people will be willing and able to purchase

Theory of Production and Theory of Production and CostsCostsFocus- mainly on the the firm. We will examine

◦Its production capacity given available resources

◦ the related costs involved

What is a firm?What is a firm?A firm is an entity concerned with the

purchase and employment of resources in the production of various goods and services.

Assumptions: ◦ the firm aims to maximize its profit with

the use of resources that are substitutable to a certain degree

◦ the firm is" a price taker in terms of the resources it uses.

What Is Production What Is Production FunctionFunctionProduction function deals with Production function deals with

the maximum output that can be the maximum output that can be produced with a limited and produced with a limited and given quantity of inputs. given quantity of inputs.

The production function is dependent on different time frames. Firms can produce for a brief or lengthy period of time.

Production FunctionProduction FunctionMathematical representation

of the relationship:Q = f (K, L, La)

Output (Q) is dependent upon the amount of capital (K), Land (L) and Labour (La) used

ASSUMPTIONS

THE PRODUCTION FUNCTIONS ARE BASED ON CERTAIN ASSUMPTIONS

1. Perfect divisibility of both inputs and outputs

2. Limited substitution of one factor for another

3. Constant technology

4. Inelastic supply of fixed factors in the short run

THE LAWS OF PRODUCTION

LAWS OF VARIABLEPROPORTIONS

LAWS OF RETURNSTO SCALE

Relates to the study of input output relationship in the short run with one variable input while other inputs are held constant

Relates to the study of input output relationship in the long run assuming all inputs to be variable

Firm’s InputsFirm’s InputsInputs - are resources that

contribute in the production of a commodity.

Most resources are lumped into three categories: ◦Land, ◦Labor,◦Capital.

Fixed vs. Variable InputsFixed vs. Variable InputsFixed inputs -resources used at a

constant amount in the production of a commodity.

Variable inputs - resources that can change in quantity depending on the level of output being produced.

The longer planning the period, the distinction between fixed and variable inputs disappears, i.e., all inputs are variable in the long run.

Production Analysis with One Variable Production Analysis with One Variable InputInput

Total product (Q) refers to the total amount of output produced in physical units (may refer to, kilograms of sugar, sacks of rice produced, etc)

The marginal product (MP) refers to the rate of change in output as an input is changed by one unit, holding all other inputs constant.

L

L

TPMP

L

Total vs. Marginal ProductTotal vs. Marginal Product

Total Product (TPx) = total amount of output produced at different levels of inputs

Marginal Product (MPx) = rate of change in output as input X is increased by one unit, ceteris paribus.

XX

TPMP

X

Production Function of a Rice Production Function of a Rice FarmerFarmer

Units of LUnits of L Total Product Total Product

(Q(QLL or TP or TPLL))Marginal Product Marginal Product

(MP(MPL)L)

00 00 --

11 22 22

22 66 44

33 1212 66

44 2020 88

55 2626 66

66 3030 44

77 3232 22

88 3232 00

99 3030 -2-2

1010 2626 -4-4

FIGURE 5.1. Total product curve. The total product curve shows the behavior of total product vis-a-vis an input (e.g., labor) used in production assuming a certain technological level.

L

QL

QL

2

6

12

20

26

30

32

Labor

Tota

l p

rod

uct

0 2 4 6 8 1097531

Marginal ProductMarginal Product

The marginal product refers to the rate of change in output as an input is changed by one unit, holding all other inputs constant.

Formula:

LL

TPMP

L

Marginal ProductMarginal ProductObserve that the marginal product

initially increases, reaches a maximum level, and beyond this point, the marginal product declines, reaches zero, and subsequently becomes negative.

The law of diminishing returns states that "as the use of an input increases (with other inputs fixed), a point will eventually be reached at which the resulting additions to output decrease"

Total and Marginal Total and Marginal ProductProduct

-10

-5

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9

TPL

MPL

Law of Diminishing Marginal Law of Diminishing Marginal ReturnsReturnsAs more and more of an input is

added (given a fixed amount of other inputs), total output may increase; however, the additions to total output will tend to diminish.

Counter-intuitive proof: if the law of diminishing returns does not hold, the world’s supply of food can be produced in a hectare of land.

Average Product (AP)Average Product (AP)

Average product is a concept commonly associated with efficiency.

The average product measures the total output per unit of input used. ◦ The "productivity" of an input is usually

expressed in terms of its average product. ◦ The greater the value of average product, the

higher the efficiency in physical terms. Formula:

LL

TPAP

L

TABLE 5.2. Average product of labor.

Labor (L)Total product of

labor (TPL)Average product of

labor (APL)

0 0 0

1 2 2

2 6 3

3 12 4

4 20 5

5 26 5.2

6 30 5

7 32 4.5

8 32 4

9 30 3.3

10 26 2.6

Rise = Y

Run = L0L

Y

The slope of the line from the origin is a measure of the AVERAGE

Y

L1 L2

a b

riseSlope =

run

Y

L

L

Q

QL

0

Total Product

a

bc

d

The average product at b is highest.

AP at c is less than at a.

AP at d is less than at c.

L

Q

TPL

Highest Slope of Line from Origin

Max APL

Inflection point

Max MPL

0 L1 L2 L3

Relationship between Average Relationship between Average and Marginal Curves: Rule of and Marginal Curves: Rule of ThumbThumbWhen the marginal is less than the

average, the average decreases.When the marginal is equal to the

average, the average does not change (it is either at maximum or minimum)

When the marginal is greater than the average, the average increases

L

AP,MP

Max APL Max MPL

0 L1L2 L3

MPL

APL

At Max AP, MP=AP

L

AP,MP

0 L1L2 L3

MPL

APL

Stage IMP>AP

AP increasing

Stage IIMP<AP

AP decreasingMP still positive

Stage IIIMP<0

AP decreasing

L

TP

0 L1 L2 L3

TPL

Three Stages of Three Stages of ProductionProductionIn Stage I

◦ APL is increasing so MP>AP.

◦All the product curves are increasing◦Stage I stops where APL reaches its

maximum at point A. ◦MP peaks and then declines at point

C and beyond, so the law of diminishing returns begins to manifest at this stage

Three Stages of Three Stages of ProductionProductionStage II

◦starts where the APL of the input begins to decline.

◦QL still continues to increase, although at a decreasing rate, and in fact reaches a maximum

◦Marginal product is continuously declining and reaches zero at point D, as additional labor inputs are employed.

Three Stages of Three Stages of ProductionProductionStage III starts where the MPL has

turned negative. ◦all product curves are decreasing. ◦total output starts falling even as the

input is increased

The Law of Variable The Law of Variable ProportionsProportions

• Elaborately stating the Law : In the short run, as the amount of variable factors increases, other things remaining equal, OP(or the returns to the factors varied will increase more than proportionally to the a amount of the variable inputs in the beginning than it may increase in the same proportion and ultimately it will increase less proportionately.

• Assuming that the firm only varies the labour (L), it alters the proportion between the fixed input and the variable input. As this altering goes on, the firm experiences the Law of Diminishing Marginal Returns.

Production ScheduleProduction ScheduleUsing the concept of MP, During the SR, under the given state of technology and other conditions remaining unchanged, with the given fixed factors, when the units of a variable factor are increased in the production function in order to increase the TP, the TP initially may rise at an increasing rate and after a point, it tends to increase at a decreasing rate because the MP of the variable factor in the beginning may tend to rise but eventually tends to diminish.

Units of Variable Input (Labour

)(n)

Total Product

(TP)

Average

Product (AP) (TPn)

Marginal Product (TPn- TPn-1)

1 20 20 20

STAGE I2 50 25 30

3 90 30 40

4 120 30 30

STAGE II

5 135 27 15

6 144 24 9

7 147 21 3

8 148 18.5 1

9 148 16.4 0

10 145 14.5 -3 STAGE III

Increasin

g

Returns

Diminishing

ReturnsNegative Returns

TP

StagesStages

Diminishing Total returns -implies reduction in total product with every additional unit of input. Diminishing Average returns -which refers to the portion of the Average Physical Product curve after its intersection with MPP curve. Diminishing Marginal returns refers to the point where the MPP curve starts to slope down and travels all the way down to the x-axis and beyond. Putting it in a chronological order, at first the marginal returns start to diminish, then the average returns, followed finally by the total returns.

ObservationsObservationsThe L of DMR becomes evident in the marginal product column.

Initially MP of Labor rises . The TP rises at an increasing rate (= MP). Average Product also rises. Stage of increasing Returns

After certain point (4th unit of Labour), the MP begins to diminish. Rate of increase in the TP slows down. Stage of diminishing returns. When AP is max, AP=MP=30 at 4th unit of labour.

AS MP diminishes, it becomes zero and negative thereafter (Stage III)

When MP is zero, TP is maximum. (148 is the highest amount of TP, when MP is equal to 0 when 9 units of labour are employed.

When MP becomes negative, TP also starts to diminish in the same proportion but AP declines after being positive up to a certain point.

QuestionQuestion

No. of fishermen

3 4 5 6 7 8 9

Daily Tuna Catch

300 450 590 665 700 725 710

Following data relates to the quantity of tuna that could be caught with different crew sizes.

Indicate the points that delineate the three stages of production .

Explanation of the stagesExplanation of the stagesThe operation of the law of

diminishing returns in three stages is attributed to two fundamental characteristics of factors of production:

i) Indivisibility of certain fixed factors.

ii) Imperfect substitutability between factors.

Marginal Revenue Marginal Revenue ProductivityProductivityThe marginal revenue productivity , also

referred to as the marginal revenue product of labor and the value of the marginal product or VMPL, is the change in total revenue earned by a firm that results from employing one more unit of labor. It determines, under some conditions, the optimal number of workers to employ at an exogenously determined market wage rate.

In a competitive firm, the marginal revenue product will equal the product of the price and the marginal product of labor. It is not efficient for a firm to pay its workers more than it will earn in profits from their labor

Total Product: Q = 30L+20L2-L3

Average Product : Q /L Marginal Product : MP = dQ/dL =

30+40L-3L2

Production function with one Production function with one variable inputvariable input

What is long run production What is long run production function ?function ?

Long run refers to that time in the Long run refers to that time in the future when all inputs are variable future when all inputs are variable inputs.inputs.

In the long run both capital and labour are included

Output can be varied by changing the Output can be varied by changing the levels of both L & K and the long run levels of both L & K and the long run production function is expressed as:production function is expressed as:

Q = f (L, K)Q = f (L, K)

THE LAW OF RETURNS TO SCALE

EXPLAINED BY

ISOQUANT CURVE

TECHNIQUE

PRODUCTIONFUNCTION

LONG RUN TOTAL PRODUCTION-LONG RUN TOTAL PRODUCTION-Returns to scaleReturns to scale

During the short period, some factors of production are relatively scarce, therefore , the proportion of the factors may be changed but not their scale. But in the long run, all factors are variable, therefore, the scale of production can be changed in the long run

Returns to scale is a factor that is studied in the long run.

Returns to scale show the responsiveness of total product when all the inputs are increased proportionately.

Returns to ScaleReturns to ScaleWhen all inputs are changed in

the same proportion (or scale of production is changed),the total product may respond in three possible ways:

1)Increasing returns to scale2)Constant returns to scale, and 3)Diminishing returns to scale

INCREASING RETURNS TO SCALEINCREASING RETURNS TO SCALEThe law of increasing returns

to scale operates when the percentage increase in the total product is more than the percentage increase in all the factor inputs employed in the same proportion.

Many economies set in and increase in return is more than increase in factors.

For e.g 10 percent increase in labour and capital causes 20 percent increase in total output. Similarly, 20 percent increase in labour and capital causes 45 percent increase in total output.

CONSTANT RETURNS TO SCALECONSTANT RETURNS TO SCALE Law of constant

returns to scale operates when a given percentage increase in the factor inputs in the same proportion causes equal percentage increase in total output.

Economies of scale are counter balanced by diseconomies of scale.

DIMINISHING RETURNS TO DIMINISHING RETURNS TO SCALESCALE The law of diminishing

returns to scale occurs when a given percentage increase in all factor inputs in equal proportion causes less than percentage increase in output.

Output increases in a smaller proportion.

Diseconomies outweigh economies of scale

Q

X,Y

IRTS Q

X,Y

DRTSQ

X,Y

CRTS

Graphically, the returns to scale concept can be illustrated using the following graphs

47

Production Isoquants/ isoquant curve/iso-product

curve• In the long run, all inputs are

variable & isoquants are used to study production decisions– An isoquant or iso-product curve is a

curve showing all possible input combinations capable of producing a given level of output

– Isoquants are downward sloping; if greater amounts of labor are used, less capital is required to produce a given output

IsoquantIsoquant

a curve showing all possible efficient combinations of input that are capable of producing a certain quantity of output

(Note: iso means same, so isoquant means same quantity)

Isoquant for 100 units of Isoquant for 100 units of outputoutput

100

Quantity of capital used per unit of time

Quantity of labor used per unit of time

K1

K2

K3

K4

L1 L2 L3 L4

100 units of output can be produced in many different ways including L1 units of labor & K1 units of capital, L2 units of labor & K2 units of capital, L3 units of labor & K3 units of capital, & L4 units of labor & K4 units of capital.

Isoquants for different output Isoquants for different output levelslevels

50

100

125



As you move in a northeasterly direction, the amount of output produced increases, along with the amount of inputs used.

It is possible for an isoquant to It is possible for an isoquant to have positively sloped sections.have positively sloped sections.



In these sections, you’re increasing the amounts of both inputs, but output is not increasing, because the marginal product of one the inputs is negative.

The lines connecting the points where the The lines connecting the points where the isoquants begin to slope upward are called isoquants begin to slope upward are called ridge lines.ridge lines.Quantity of capital used per unit of time


ridge lines

No profit-maximizing firm will operate at a No profit-maximizing firm will operate at a point outside the ridge lines, since it can point outside the ridge lines, since it can produce the same output with less of both produce the same output with less of both outputs.outputs.



L1 L2

K2

K1

Notice, for example, that since points A & B are on the same isoquant, they produce the same amount of output.

However, point B is a more expensive way to produce since it uses more capital & more labor.

B

A

Marginal rate of technical Marginal rate of technical substitution (MRTS)substitution (MRTS)

The slope of the isoquant

The rate at which you can trade off inputs and still produce the same amount of output.

For example, if you can decrease the amount of capital by 1 unit while increasing the amount of labor by 3 units, & still produce the same amount of output, the marginal rate of technical substitution is 1/3.

Marginal Rate of Technical Marginal Rate of Technical Substitution (MRTS)Substitution (MRTS)or slope of an isoquantor slope of an isoquant

ΔK/ΔL = - MPL/MPK

the negative of the ratio of the marginal products of the inputs, with the input on the horizontal axis in the

numerator.

Other types of IsoquantsOther types of IsoquantsLinear IsoquantsL- shaped IsoquantsKinked Isoquants

How does output respond to changes How does output respond to changes in scale in the long run?in scale in the long run?

Three possibilities:1. Constant returns to scale 2. Increasing returns to scale3. Decreasing returns to scale

Constant returns to scale Constant returns to scale

Doubling inputs results in double the output.

Constant returns to scaleConstant returns to scaleAttributed to the limits of the

economies of scale.When economies of scale reach

their limits and diseconomies of scale are yet to begin, returns to scale become constant.

Increasing returns to scale Increasing returns to scale

Doubling inputs results in more than double the output.One reason this may occur is that a firm may be able to use production techniques that it could not use in a smaller operation.

Decreasing returns to scale Decreasing returns to scale

Doubling inputs results in less than double the output.One reason this may occur is the difficulty in coordinating large organizations (more paper work, red tape, etc.)

Graphs of Constant, Increasing, & Graphs of Constant, Increasing, & Decreasing Returns to ScaleDecreasing Returns to Scale

Constant Returns to Scale: isoquants for output levels 50, 100, 150, etc. are evenly spaced.

Capital

Labor

150

100

50

150

Capital

Labor

50100

150

Capital

Labor

10050

Increasing Returns to Scale: isoquants for output levels 50, 100, 150, etc. get closer & closer together.

Decreasing Returns to Scale: isoquants for output levels 50, 100, 150, etc. become more widely spaced.

Iq4

Iq3

Iq2

Iq1 = 100

A

B

C

D

K4

K3

K2

K1

0L1 L2 L3 L4

Units of L

Unit

s of

K

= 200

= 300

= 400

ISOQUANT MAP- A family or a group of isoquants is called an ISOQUANT MAP

66

The Isocost Line

0 1 2 3 4 5 6 7 8 9 10 Labor, L (worker-hours employed)

Cap

ital,

K (

mac

hine

s re

nted

)

Cost = Rs50Per unit price of labor input = Rs10/hourPer unit price of capital input = Rs5/machine

2

4

8

10

a

b

c

d

e

f

6

A

B

M=PL.QL+PK.QK

Where, M=total outlay PL= price per unit of labor PK= price per unit of capital QL= units of labor QK= units of capital

Slope of isocost line= OA/OF

Slope of isocost line can be changed in two ways:1) Change in the factor price, and2) Change in total outlay or total cost

Slope of isocost line

input capital ofunit per price

inputlabour ofunit per price

68

Changes in One factor Price

Decrease in the factor price causes rightward shift and increase in factor price causes leftward

shift in iso-cost line.

0 1 2 3 4 5 6 7 8 9 10

Cap

ital,

K (

mac

hine

s re

nted

)

2

4

6

8

10

Labor, L (worker-hours employed)

a

f

Cost = 500; labor,R = 16.5 or 10or 1/ hourThe money wage, W = Rs5/machine

…Rs10

…Rs1

A Change in unit price of labor

hRs16.5

69

L

K

Units of labour(w)

Slope = -w/r

Direction of increasein total cost

Change in total outlay or total cost

TC=Rs. 50

TC= Rs. 75

TC= Rs. 100

70

L

K

Units of Labour

•

Isoquants and Cost Minimization

•

Q=200

Q=300

Q=100

IQ 2IQ 3

TC=Rs50

•

•

0 2 4 6 8 10 12 14 16 18 20

TC=Rs=75

TC=Rs100

P

P”

P’

•N

M

71

Optimization & Cost• Expansion path gives the efficient

(least-cost) input combinations of labor and capital needed for every level of output.Derived for a specific set of input pricesAlong expansion path, input-price ratio

is constant & equal to the marginal rate of technical substitution

• It is defined as the locus of tangency points between iso-cost lines and isoquants.

•It implies to Long run because: No input is fixed.Path starts from origin indicating that if output is zero costs are zero.•Expansion path gives us thelevel of output & one leastcombination that canproduce this level of output.•Movement along the line givesthe costs at which output canbe expanded•So called Expansion Path.

EXPANSION PATH

Labor input

Capita

l input

Estimation of production function – Cobb Estimation of production function – Cobb Douglas Production FunctionDouglas Production Function

The function used to model production is of the form:

Q(L,K) = ALaKb

where:Q = total production L = labor inputK = capital inputA = total factor productivity a and b are the output elasticities of labor and

capital, respectively. These values are constants determined by available technology.

Output elasticity measures the responsiveness of output to a change in levels of either labor or capital used in production, ceteris paribus. E.g. if a= 0.15, a 1% increase in labor would lead to approximately a 0.15% increase in output.

Total Factor productivity :TFP tries to assess the efficiency with which both capital and labour are used. Once a country's labour force stops growing and an increasing capital stock causes the return on new investment to decline, TFP becomes the main source of future economic growth. It is calculated as the percentage increase in output that is not accounted for by changes in the volume of inputs of capital and labour. So if the capital stock and the workforce both rise by 2% and output rises by 3%, TFP goes up by 1%.

Returns to scale based on Returns to scale based on Cobb Douglas functionCobb Douglas functionIf a+b = 1,the production function has

constant returns to scale (CRTS). That is, if L and K are each increased by 20%, then Q increases by 20%.

If output increases by less than that proportional change, there are decreasing returns to scale (DRS). i.e. a+b<1

If output increases by more than that proportion, there are increasing returns to scale (IRS) ). i.e. a+b>1

Leontif Production Leontif Production functionfunctionCapital and labor are perfect

complements. Capital and labor are used in fixed-

proportions. Q = min {bK, cL} Since capital and labor are

consumed in fixed proportions there is no input substitution along isoquants (hence, no MRTSKL).

production function

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