Production AnalysisProduction Analysis

Producer has to decide on…..Producer has to decide on…..

How much to produceHow much to produce What capacity to be installedWhat capacity to be installed What combination of inputs to be What combination of inputs to be

employed to maximise profits and employed to maximise profits and minimise costsminimise costs

At what price to sellAt what price to sell

Production functionProduction function A production function is a functional specification that provides A production function is a functional specification that provides

the most efficient combination of input with which a chosen the most efficient combination of input with which a chosen target level of output can be produced. target level of output can be produced.

It is specific to each industry and technology.It is specific to each industry and technology.

Decisions that producers need to make:Decisions that producers need to make:– To meet increased demand, should the firm go in for capacity To meet increased demand, should the firm go in for capacity

expansion or stretch the existing production facilities.expansion or stretch the existing production facilities.– How should they handle existing idle capacity.How should they handle existing idle capacity.

Production FunctionProduction Function Decision variable involved in production Decision variable involved in production

decisions are – Inputs and Outputdecisions are – Inputs and Output Input is anything that the firm employs in the Input is anything that the firm employs in the

production processproduction process Output is what the firm produces making use Output is what the firm produces making use

of inputs.of inputs. Production functions change when the Production functions change when the

technical process of production change technical process of production change leading to an entirely different set of input leading to an entirely different set of input combinations related in an entirely different combinations related in an entirely different manner.manner.

Production Function with two Production Function with two variable inputsvariable inputs

Q = f (K , L) where K is capital and L Q = f (K , L) where K is capital and L is labour.is labour.

Given a target level of output, this Given a target level of output, this function gives us the highest level of function gives us the highest level of output that can be produced from a output that can be produced from a given combination of inputs.given combination of inputs.

Production function can take many Production function can take many forms like F(L,K) = 3L + 2Kforms like F(L,K) = 3L + 2K2, 2, or, 5K or, 5K0.5 0.5

LL0.5,0.5, or any other form or any other form

Production function…Production function…

Consider the following combination of Consider the following combination of inputs for the production of a given level inputs for the production of a given level of output, say 160 cars.of output, say 160 cars.

Given f(L,K) = 5KGiven f(L,K) = 5K0.50.5LL0.50.5

Combination L K Output

A 50 20.5 160

B 40 25.6 160

C 30 34.13 160

D 20 51.2 160

Production function…Production function…

Consider the following combination of Consider the following combination of inputs for the production of different levels inputs for the production of different levels of outputof output

Given f(L,K) = 5KGiven f(L,K) = 5K0.50.5LL0.50.5

Combination L K Output

A 50 20.5 160

B 40 40 200

C 30 120 300

D 20 128 400

IsoquantIsoquant

An isoquant is a curve on which An isoquant is a curve on which every point satisfies the production every point satisfies the production function and thus, all combination of function and thus, all combination of L and K on an isoquant are L and K on an isoquant are technically efficient combination with technically efficient combination with which the given level of output can which the given level of output can be produced.be produced.

Each isoquant corresponds to a Each isoquant corresponds to a different level of output.different level of output.

Production Function with Production Function with Isoquant MapIsoquant Map

Y O U T P U T

8 37 60 83 96 107 117 127

7 42 64 78 90 101 110 119

6 37 52 64 73 82 90 97

5 31 47 58 67 75 82 89

4 24 39 52 60 67 73 79

3 17 29 41 52 58 64 69

2 8 18 29 39 47 52 65

1 4 8 14 20 27 24 21

X 1 2 3 4 5 6 7

The preceding table represents a production The preceding table represents a production function with two inputs, X and Yfunction with two inputs, X and Y

It can be observed that combinations (2,6), It can be observed that combinations (2,6), (3,4), (4,3), (6,2)yield same level of output, (3,4), (4,3), (6,2)yield same level of output, that is 52.that is 52.

By connecting the combinations we get the By connecting the combinations we get the isoquant corresponding to output level 52isoquant corresponding to output level 52

Similar combinations for different levels of Similar combinations for different levels of output can be produced can be extracted output can be produced can be extracted from the tablefrom the table..

Isoquant mapIsoquant map

IsoquantsIsoquants Graphical representation of production functionGraphical representation of production function A curve drawn through the technically feasible combinations of inputs to A curve drawn through the technically feasible combinations of inputs to

produce a target level of outputproduce a target level of output

Marginal rate of technical substitution is the substitution of one input for Marginal rate of technical substitution is the substitution of one input for another. MRTS = -change in K/change in laboranother. MRTS = -change in K/change in labor

Output 160Output 200

Output 260

K

L

Properties of IsoquantsProperties of Isoquants They are downward sloping – That is as you employ They are downward sloping – That is as you employ

more and more of the input on the X axis, you more and more of the input on the X axis, you necessarily employ less of the input on the Y axis in necessarily employ less of the input on the Y axis in order to maintain the same level of output. Employing order to maintain the same level of output. Employing more of both inputs would lead to a higher isoquant.more of both inputs would lead to a higher isoquant.

They are convex to the origin – This happens as the They are convex to the origin – This happens as the power to substitute diminishes, called as the power to substitute diminishes, called as the marginal physical product as we employ more and marginal physical product as we employ more and more of a factor. However, in case of perfect more of a factor. However, in case of perfect substitutes the isoquant is a downward sloping substitutes the isoquant is a downward sloping straight line with a constant slope, while for perfect straight line with a constant slope, while for perfect complements the isoquant is a L shaped curve.complements the isoquant is a L shaped curve.

Properties of IsoquantsProperties of Isoquants They do not intersect each otherThey do not intersect each other

Application of IsoquantsApplication of Isoquants Isoquants enable the decision maker to Isoquants enable the decision maker to

conceptualize the trade-offs involved in conceptualize the trade-offs involved in substitution between inputs. Managers consider substitution between inputs. Managers consider the costs and benefits of substituting one input the costs and benefits of substituting one input for another and select that one where the net for another and select that one where the net benefits are maximised.benefits are maximised.

Isoquant model helps the decision maker to Isoquant model helps the decision maker to figure out the increase /decrease in output with a figure out the increase /decrease in output with a change in input.change in input.

Production function with one variable inputProduction function with one variable input

Short run is defined as a period during which Short run is defined as a period during which only one of the inputs can be varied.only one of the inputs can be varied.

Long run is defined as a period during which no Long run is defined as a period during which no factor is fixed and all input factors can be varied.factor is fixed and all input factors can be varied.

Average Product : Q /L Average Product : Q /L

Marginal Product : MP = dQ/dL ie it is the Marginal Product : MP = dQ/dL ie it is the increase in output for a unit increase in the increase in output for a unit increase in the variable input.variable input.

Production functionProduction function Law of diminishing returns or the law of Law of diminishing returns or the law of

variable proportions - According to this variable proportions - According to this relationship, in a production system with fixed relationship, in a production system with fixed and variable inputs (say factory size andand variable inputs (say factory size and labor), beyond some point, each additional unit of ), beyond some point, each additional unit of variable input yields less and less output variable input yields less and less output

Increasing return is the stage where with Increasing return is the stage where with each additional unit of variable input each additional unit of variable input employed, the marginal product increases.employed, the marginal product increases.

12

3

1 – Stage of increasing returns; Ex>12 – Stage of decreasing returns,0<Ex<1

3 – Stage of negative returns; Ex<0

OUTPUT

LABOR

Production functionProduction function Diminishing return is the stage where Diminishing return is the stage where

with each additional unit of variable with each additional unit of variable input employed, the output increases input employed, the output increases but at a decreasing rate.but at a decreasing rate.

The stage where with increase in The stage where with increase in variable input, the decreasing marginal variable input, the decreasing marginal product becomes negative, resulting in a product becomes negative, resulting in a decline of total output. It is the stage of decline of total output. It is the stage of negative returns. At this point the negative returns. At this point the variable factor becomes counter variable factor becomes counter productive.productive.

12

3

1 – Stage of increasing returns; Ex>12 – Stage of decreasing returns,0<Ex<1

3 – Stage of negative returns; Ex<0

OUTPUT

LABOR

Total Product: Q = 30L+20LTotal Product: Q = 30L+20L22-L-L33

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

30+40L-3L30+40L-3L22

Production function with one Production function with one variable inputvariable input

Production ElasticityProduction Elasticity Production elasticity is the proportionate Production elasticity is the proportionate

change in output due to a proportionate change in output due to a proportionate change in input.change in input.

∆∆Q / ∆X * X / Q = MPx * 1 / APxQ / ∆X * X / Q = MPx * 1 / APx Production elasticity greater than one Production elasticity greater than one

indicates that output increases by a indicates that output increases by a proportion greater than the increase in proportion greater than the increase in input. input.

In cases where the elasticity is zero, there is In cases where the elasticity is zero, there is no change in output due to a change in no change in output due to a change in input.input.

For a value of production elasticity less than For a value of production elasticity less than zero indicates that output decreases with a zero indicates that output decreases with a given increase in input.given increase in input.

Three Stages of ProductionThree Stages of Production

Stage 1: AP is increasing, MP is Stage 1: AP is increasing, MP is increasing and Production Elasticity is increasing and Production Elasticity is > 1. This stage corresponds to output > 1. This stage corresponds to output levels that indicate underutilization of levels that indicate underutilization of capacity.capacity.

Stage 2: AP and MP are decreasing, Stage 2: AP and MP are decreasing, until MP=0 and Production Elasticity until MP=0 and Production Elasticity is 0 < Prod.Elas < 1. Producer’s is 0 < Prod.Elas < 1. Producer’s optimal choice of employment of optimal choice of employment of variable input lies in this stage. variable input lies in this stage.

Three Stages of ProductionThree Stages of Production

Stage 3: MP and AP continue to Stage 3: MP and AP continue to decrease and MP <0; leading to decrease and MP <0; leading to decrease in total product with decrease in total product with increasing units of input and increasing units of input and Production Elasticity < 0. No rational Production Elasticity < 0. No rational producer would want to be in this producer would want to be in this stage. This stage corresponds to stage. This stage corresponds to output levels that indicate output levels that indicate overutilization of capacity.overutilization of capacity.

Relationship between Marginal physical Relationship between Marginal physical product and marginal rate of technical product and marginal rate of technical

substitutionsubstitution MRTS is equal to slope of the MRTS is equal to slope of the

isoquant ie =-isoquant ie =-∆K/ ∆L∆K/ ∆L MRTS can also be expressed MRTS can also be expressed

algebraically. While moving from A to algebraically. While moving from A to B or B to C on an isoquant, the B or B to C on an isoquant, the following condition is to be satisfied:following condition is to be satisfied:

MPMPll x ∆L + MP x ∆L + MPkk x ∆K = 0 x ∆K = 0 Therefore - ∆K/ ∆L = MPTherefore - ∆K/ ∆L = MPll/MP/MPkk

Returns to ScaleReturns to Scale The rate at which output increases to a proportionate The rate at which output increases to a proportionate

change in all inputs is known as the degree of returns change in all inputs is known as the degree of returns to scale. Increasing output given constraints of to scale. Increasing output given constraints of technology means moving from one isoquant to technology means moving from one isoquant to another.another.

Increasing Returns to Scale - When output increases Increasing Returns to Scale - When output increases by a proportion greater than the proportionate by a proportion greater than the proportionate increase in all inputs. Specialization and division of increase in all inputs. Specialization and division of labor assists in being in this stage.labor assists in being in this stage.

Decreasing returns to scale – Output increases by a Decreasing returns to scale – Output increases by a proportion less than that of the increase in inputs.proportion less than that of the increase in inputs.

Constant returns to scale – Output increases by the Constant returns to scale – Output increases by the same proportion as the increase in inputs.same proportion as the increase in inputs.

Estimation of production functionsEstimation of production functions

There are a variety of production functionsThere are a variety of production functions Linear production functions: Q = a + bX, Linear production functions: Q = a + bX,

these are subject to constant returns onlythese are subject to constant returns only Quadratic prod. Functions: Q = a + bX – Quadratic prod. Functions: Q = a + bX –

cXcX2 2 , these capture the diminishing returns , these capture the diminishing returns phase ( in –cXphase ( in –cX2 2 ) but not the increasing ) but not the increasing returns to scale.returns to scale.

Cubic form pfs.: Q = a + bX + cXCubic form pfs.: Q = a + bX + cX2 2 -dX -dX3 3 , , these capture both the increasing and the these capture both the increasing and the diminishing returns to scalediminishing returns to scale

Power function : Q =aXPower function : Q =aXbb

Cobb – Douglas Production Cobb – Douglas Production FunctionFunction

Q = A LQ = A Lαα K Kββ

Q = total production (the monetary value of Q = total production (the monetary value of all goods produced in a year); all goods produced in a year); LL = = labor input ;input ;KK = = capital input ; input ; AA = = total factor productivity; α and β are the output ; α and β are the output elasticities of labor and capital, respectivelyelasticities of labor and capital, respectively

Where, Where, αα + + ββ indicates Returns to Scale indicates Returns to Scale

If > 1, it exhibits Increasing Returns to ScaleIf > 1, it exhibits Increasing Returns to Scale

If < 1, it exhibits Decreasing Returns to ScaleIf < 1, it exhibits Decreasing Returns to Scale

If = 1, it exhibits Constant Returns to ScaleIf = 1, it exhibits Constant Returns to Scale

Optimal Input LevelsOptimal Input Levels Level of output determines the demand Level of output determines the demand

for inputs and employment of inputs is for inputs and employment of inputs is related to revenue generated from the related to revenue generated from the output.output.

Stage of decreasing – diminishing Stage of decreasing – diminishing returns is where the producer should be returns is where the producer should be operating with one variable input.operating with one variable input.

Total revenue product is total product Total revenue product is total product multiplied by the market price of outputmultiplied by the market price of output

Optimal Input LevelsOptimal Input Levels Marginal revenue product is the change in Marginal revenue product is the change in

total revenue product due to a unit total revenue product due to a unit change in the variable input = MP * Pricechange in the variable input = MP * Price

Total variable cost is total cost of the Total variable cost is total cost of the variable input and is arrived at by variable input and is arrived at by multiplying quantity of variable input by multiplying quantity of variable input by its price.its price.

Marginal variable cost is the change in Marginal variable cost is the change in total variable cost due to a unit change in total variable cost due to a unit change in the variable input.the variable input.

Optimal Input LevelsOptimal Input Levels

With one variable input, producer With one variable input, producer produces in stage of decreasing returns produces in stage of decreasing returns produces upto the point where produces upto the point where Marginal Marginal Revenue Product (MRP) = Marginal Revenue Product (MRP) = Marginal Variable Cost.Variable Cost.

MRP = MP * MR=MVCMRP = MP * MR=MVC With many variable inputs:With many variable inputs:

MPMPLL / P / PLL = MP = MPKK / P / PK K = ……….= ……….

Optimal Combination of InputsOptimal Combination of Inputs Given that each combination of Given that each combination of

inputs have a different cost attached, inputs have a different cost attached, the producer needs to arrive at the the producer needs to arrive at the least cost combination. The concept least cost combination. The concept of the ‘isocost’ line is used for of the ‘isocost’ line is used for arriving at the optimal combination.arriving at the optimal combination.

An isocost line gives the combination An isocost line gives the combination of inputs that can be purchased in of inputs that can be purchased in the market at the going market the market at the going market prices with a tentative budget. All prices with a tentative budget. All combinations on this line result in the combinations on this line result in the same total cost.same total cost.

Optimal Combination of InputsOptimal Combination of Inputs

The point of tangency between an isocost The point of tangency between an isocost line and an isoquant gives the least cost line and an isoquant gives the least cost combination of inputs with which the combination of inputs with which the output level corresponding to the isoquant output level corresponding to the isoquant can be produced. Isocost line is linear as can be produced. Isocost line is linear as input prices are constant.input prices are constant.

For a given target level of output, the For a given target level of output, the combination of inputs on the isocost line, combination of inputs on the isocost line, which is tangential to the isoquant is the which is tangential to the isoquant is the optimal combination. The optimum point optimal combination. The optimum point must be technically efficient (lie on the must be technically efficient (lie on the relevant isoquant) and has to satisfy the relevant isoquant) and has to satisfy the cost constraint). cost constraint).

Optimal Combination of InputsOptimal Combination of Inputs Isoproduct curve : This curve gives us the Isoproduct curve : This curve gives us the

combination of outputs that two levels of constant combination of outputs that two levels of constant input quantities can produce, assuming production is input quantities can produce, assuming production is most efficient and the inputs are completely most efficient and the inputs are completely exhausted. The curve is concave assuming exhausted. The curve is concave assuming diminishing returns occur when specialization diminishing returns occur when specialization increases. This curve is also called the increases. This curve is also called the production production frontier frontier as it indicates the highest levels of the two as it indicates the highest levels of the two outputs that can be produced with the given inputs.outputs that can be produced with the given inputs.

The economy can go beyond the frontier only when The economy can go beyond the frontier only when the frontier shifts outward. This happens when:the frontier shifts outward. This happens when:

More is produced as the resource base has expanded More is produced as the resource base has expanded or technological change has occurred.or technological change has occurred.

If there is trade between economies, then it is If there is trade between economies, then it is possible for an economy to consume more than it has possible for an economy to consume more than it has produced.produced.