FMG 24: Theory of Production

Managerial Economics

Dr. Subhasis Bera

Your mother makes delicious Puri and Sabji.

Your friends are mainly visit your house to

have Puri and Sabji. How your mother

manages kitchen when there are five

friends as guest and how does she changes

of doing things at kitchen when there are

twenty five of your friends are visiting your

house just to have Puri and Sabji?

What to produce and how to produce?

How does she manages her resources ( raw material as well as labor help) to prepare sufficient amount of Puri Sabji without wasting anything

As a manager once you know the demand for a particular product which has utility,

next thing that you need to consider is how to create that utility.

This creation of utility is possible by any of three following way

i) by change in the form of goods

ii) Change in the temporal form of goods

iii) Increase in the quantity of goods

What to produce and how to produce?

what to produce and how to produce?

A manager must know the volume of the factor inputs to be used to produce a certain amount of output

Creation of utility is known as the production

From the demand analysis a manager can decide what to produce

Production is a process of transforming the inputs into goods and services.

There are numbers of inputs required to produce a particular types of good or

services and therefore production function can be represented as

The production function refers to the physical relationship between the inputs

or resources of a firm and their output of goods and services at a given period of

time, ceteris paribus.

Here technology determines the type, quantity and proportion of inputs.

Technology(T) also determines the maximum limit of total output from a given

combinations of inputs. We need to remember that at a particular point of time T

is constant and therefore T is included only in the long run production functions.

(L, K,T,E,O,....)q f

Introduction

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

Short run is a time period when at least one factor input remain constant.

In the long run all the factor inputs are variable.

Therefore SR production function is

From this production function we can calculate Average Product and Marginalproduct of any factor input.

From the above we can get

and

(L, K)q f

L

TP qAP

L L

( )L

TP qMP

L L

Operational Time Period: Short Run and Long Run

From the production function it is obvious that changes in the input will change the

volume of output. In SR changes in the output due to the change input follows law of

variable proportion. Law of variable proportion states that with increase in the

quantity of the variable factor, its marginal and average products will eventually

decline while other inputs remain unchanged. This law of variable proportion also

known as law of diminishing returns.

L

K

TP

Short run Properties: Law of Variable Proportion

Note: this law never say when it will start to take effect. It assumes that all the inputs have the same productivity which is not true

You can think of you new mobile or new laptop or tablet to understand

law of diminishing returns

In SR technology is fixed. Therefore change in technology have impact on the

production function in the long run.

Since LR is a sum total time of number of SR therefore a change in technology will

only shift the production function, ceteris Paribas.

TP

TP1

TP3

TP2

L

Impact of Technology

Labor0

Total Q

TP

L3

Maximum MP

L1 L2

A

C

B

AP and MP of Short Run Production Function

A manager must keep this relation in mind for future

reference

Slope of Tangent at any point measures MP of variable factor input

Slope of the line joining origin and the point

measures AP

L

AP,MP

L1L2 L3

MPL

Stage IMP>AP

AP increasing

Stage IIMP<AP

AP decreasingMP still positive

Stage IIIMP<0

AP decreasing

L

TP

L1 L2 L3

TPL

APL

Three Stages of Production

Stage I: from zero units of the variable input to where AP is maximized (where MP = AP)

Stage II: from the maximum AP to where MP = 0

Stage III: from where MP = 0

Increase in the volume of one factor input increases the efficiency of use of fixed factor

and therefore TP increases at increasing rate. This is known as first stage of production.

At this stage both AP and MP increase.

In the second stage, after the efficient use of fixed input, increase in variable input

increases the production at a decreasing rate because of the law of diminishing return.

At this stage MP is falling but still MP > AP.

In the third stage of production increase in the variable input reduces the volume of

production i.e., it gives the negative return and here both AP and MP fall and MP < AP.

Three Stages of Production

It is obvious that none of the firm will operate at stage III.

None will operate at stage-I as they will continue increasing variable input as long as AP and MP increase.

To determine the optimum level of factor inputs we need to know the MRP (marginal

revenue product ) and the cost of labour.

Where, MRPL=change in total revenue due to a change in the volume of labour input used

And MLC= change in the total labor cost due to a unit change in the volume of labour

engaged in the production process.

Now if total labour cost = w. L, then MLC = w

Therefore from the profit maximizing condition (MRP = MLC) we can have that firm will

employ the variable factor input in such a way so that

Therefore in case of multiple variable factor input optimum level will be derived from the

condition

L L

(TR) (TR)MRP = = =MR. MP

d d dQ

dL dQ dL

1 2

1 2

k

k

MP MP MP

w w w

MR.MPL w MR / MPLw

Short Run Equilibrium: Optimum level of Single Output

In LR all the factor inputs are Variable.

There are various combinations that yields the same level of output.

Locus of combinations of two factor inputs that yields same level of out is called

isoquant.

2( , )Q L K Q

L

K

A

B

C

K1

L1 L3

K3

K2

L2

1( , )Q L K Q

Properties of IQ

1) Downward slopping from left to right

2) Convex to origin

3) Two IQ never intersect each other

Long Run Production Function

Its the amount of one factor input that must be substituted for 1 unit of another input

to maintain a constant level of output.

Slope of Isoquant =

MRTSLK diminishes as amount of substitution increases.

LLK

K

MPMRTS

MP

L per period

K per period

q = 20

LA

KA

KB

LB

A

B

Marginal Rate of Technical Substitution (MRTS)

Depending on the factor input substitutability we can have various types ofisoquant.

1) Linear Isoquant

2) Right-Angle Isoquant or Input-Output Isoquant

3) Convex Isoquant

Linear Isoquant- In Linear Isoquants there is perfect substitutability of inputs.

e.g., In a power plant equipped to burn oil or Gas. Various amounts of electricitycould be produced by burning Gas, Oil or a combination. i.e., Oil and Gas areperfect substitutes.

Input-Output Isoquant- in this case factor inputs are not substitute.

Convex isoquant- in this case factor input Substitutability are limited

Factor Input Substitutability

K

L

1. a firm will continue to employ a

factor input unless its MP

becomes negative. Therefore

threshold level for employing a

factor unit is MPx = 0

2. Ridgeline is the locus of points

on the isoquants where MPx is

zero.

3. Firm will substitute factor inputs

within the ridgeline.

4. the isoquant in between the

ridge line known as economic

region where input substitution

is economically viable.

Ridgelines: Economic Region

MPL =0

MPK =0

In Long Run if all the factors are changed in same proportion then new can say that

there is a change in the scale of production.

Changes the volume of output due to Proportional change in the input is known

Return to Scale (RTS).

Depending on the changes in the volume of output we can have IRS, DRS and CRS.

Return to Scale

Consider a 450 line which shows the factor input ratio remain same on the line.

Distance between a and b = that of b and c → CRS

Distance between a and b < that of b and c → DRS

Distance between a and b > that of b and c → IRS

L per period

K per period

q = 3

q = 2

q = 1

a

b

cq = 3

c

q = 3

c

Return to Scale: IRS, DRS, CRS

Returns to scale can also be described using the following equation

hQ = f(kX, kY)

if h > k then IRS

if h = k then CRS

if h < k then DRS

Many production system exhibit first increasing, then constant and then decreasingreturn to scale. The region of increasing returns is attributable to specialization. Asoutput increases, specialized labour can be used and efficient large-scalemachinery can be used in the production process.

Beyond some scale of operation, however further gains from specialization arelimited and coordination problems emerge.

When coordination expenses more than offset additional benefits of specialization,decreasing returns to scale set in.

Return to Scale: Mathematical

Isocost lines shows the combination of two factor inputs that a producer can

purchase at given prices and income.

Equation of the isocost line is M= PL .L + PK .K

K

L

K1

L1L2 L3

K2

Isocost Line

K

L

1L

AB

C

W1W2

W1

1K

1Iq

2Iq

3Iq2K

2L

Equilibrium condition is

Slope of Isoquant = Slope of Isocost Line

Producer’s Equilibrium

At equilibrium producer maximises its profit

Now we assume that our production function is Q=f (L,K)

Isocost line is PL .L + PK . K =Y

Objective of a manager is to make maximum profit out of the production and there

to maximise the gap between income and expenditure.

There are two ways manager can maximise profit-

i) by maximising output keeping the cost constant

ii) by minimising cost keeping the output level fixed.

Producer’s Equilibrium: Manager’s Problem

In this case our problem is

Therefore from FOC we get

and

and

Now solving (i) and (ii) we get

i.e., optimal input proportion require that the ratio of marginal product to pricemust be equal for all input factors.

Max M= ( , ) ( L-P K) L Kf L K Y P

( , )0..................( )L

M f L KP i

L L

( , )0................( )K

M f L KP ii

K K

0............( )L K

MY P L P K iii

L K

L K

MP MP

P P

Producer’s Equilibrium: Output Maximization

In this case our problem is

Therefore from FOC we get

and

and

Now solving (i) and (ii) we get

i.e., optimal input proportion require that the ratio of marginal product to price

must be equal for all input factors.

( , ). 0..................( )L

M f L KP i

L L

( , ). 0................( )K

M f L KP ii

K K

* ( , ) 0............( )M

Q f L K iii

L K

L K

MP MP

P P

[ * ( , )]L KM P L P K Q f L K

Producer’s Equilibrium: Cost Minimization

There are various types of production function.

i) Cubic form ,

where a, b, c are coefficients

ii) Quadratic function

iii) Power function

A very widely used production function is Cobb-Douglas Production function andrepresented as

Where, a = Total factor productivity

α, β are output elasticities.

2 3Q a bL cL dL

2Q a bL cL

bQ aL

Q aL K

Various Types of Production Function

01. For Q to be a positive number both the factor must exits i.e., if either of the factor vanishesthen so the output.

02. This production function exhibit increasing, decreasing and constant return to scale.Originally CD assumes that returns to scales are constant.

(α+ β ) represents the return to scale .

If (α+ β ) > 1 the there is IRS

If (α+ β ) = 1 then there is CRS

And if (α+ β ) < 1 then there is DRS

03. This allows calculating MP of any factor input keeping other constant and there this is usefulfor short run analysis also.

04. Parameters can be estimated using linear regression model.

05. A Theoretical production function assumes technology is constant. However the data fittedby the managers may span a period over which technology has progressed. One of theindependent variables in the previous equation could represent technological change and thusadjust the function to take technology into consideration.

06. it has attractive mathematical characteristics, such as diminishing marginal returns to eitherfactor of production.

Cobb-Doglus production Function: Properties q aL K

Cobb-Doglus production Function: Limitation

The Cobb-Douglas production function was not developed on the basis of any knowledge of engineering, technology, or management of the production process.

Neither Cobb nor Douglas provided any theoretical reason why the coefficients α and β should be constant over time or be the same between sectors of the economy. Remember that the nature of the machinery and other capital goods (the K) differs between time-periods and according to what is being produced. So do the skills of labor (the L).

This does not allow to calculate MP going through all three stages of production in one specification ( a cubic function would be necessary to achieve this).

This can not show a firm or industry passing through, increasing, constant and decreasing returns to scale.

There are also problem of specification of data to be used in empirical estimates.

q aL K

This is an extreme case of perfect complements of the factor inputs. This

production function has L shaped isoquant.

Production function represented as

and if then

and K is considered to be binding constraint in the production process.

min ,L K

Q

L K

KQ

Leontief Production Function

A special type of production function ( introduced by Arrow, Chenery, Minhas and

Solow and known as ACMS function) represented as

Where A = efficiency parameter

α = Distribution parameter that shows the relative factor shares

ρ = substitution parameter which help us to derive elasticity of substitution

r = scale parameter which determine the degree of homogeneity

(1 )r

Q A K L

CES Production Function

After formulating the model, one can estimate the production function.

There are mainly three methods of estimating production function

Estimation Production Function

Time series Analysis

Cross Sectional Analysis

Engineering Analysis

Now collect data from the Prowess Database and estimate the production function for metal industry

for the period 2000-2015

This is an Assignment

Thank You!

She is also producing something as per the demand of her children. Short of resources does not allow her to meet the demand and she is identified as poor one