Ch-4 Production Theory

8/6/2019 Ch-4 Production Theory

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Ch 4 THE THEORY OFPRODUCTION

Production theory forms thefoundation for the theory of supply

Managerial decision making involvesfour types of production decisions:

1. Whether to produce or to shut

down

2. How much output to produce

3. What input combination to use

4. What type of technology to use

Ch 5

Ch 4



Production involves transformation

of inputs such as capital,

equipment, labor, and land into

output - goods and services

In this production process, the

manager is concerned with

efficiency in the use of the inputs

- technical vs. economical

efficiency



Two Concepts of Efficiency

Economic efficiency:

occurs when the cost of producing

a given output is as low as possible

Technological efficiency:

occurs when it is not possible to

increase output without increasing

inputs



The objective of efficiency willprovide us with some basic rules

about the manner in which firms

should utilize inputs to produce

goods and services



You will see that basic productiontheory is simply an application of

constrained optimization:the firm attempts either to minimizethe cost of producing a given levelof output

or

to maximize the output attainablewith a given level of cost.

Both optimization problems lead tosame rule for the allocation of inputs and choice of technology



Production Function

A production function is a table or amathematical equation showing themaximum amount of output that can beproduced from any specified set of inputs,

given the existing technology

f 2(x)f

1(x)

f 0(x)

x

Q Improvement of technology

f 0

(x) - f 2

(x)

Q = output

x = inputs



Production Function continued

Q = f(X1, X

2, …, X

k)

where

Q = output

X1, …, X

k= inputs

For our current analysis, let’s reduce theinputs to two, capital (K) and labor (L):

Q = f(L, K)



Production Table

Units of K Employed Output Quantity (Q)

8 37 60 83 96 107 117 127 128

7 42 64 78 90 101 110 119 120

6 37 52 64 73 82 90 97 104

5 31 47 58 67 75 82 89 954 24 39 52 60 67 73 79 85

3 17 29 41 52 58 64 69 73

2 8 18 29 39 47 52 56 52

1 4 8 14 20 27 24 21 17

1 2 3 4 5 6 7 8

Units of L Employed

Same Q can be produced with differentcombinations of inputs, e.g. inputs are substitutable

in some degree



All of these outputs are assumed to

be technically efficient

But which one is economically efficient?

That is the question facing the DM



Types of Production Functions

Production function with one variable

input

Production function with two variable

inputs Production function with all variable

inputs



Short-Run and Long-Run Production

In the short run some inputs are

fixed and some variable

e.g. the firm may be able to vary

the amount of labor, but cannotchange the amount of capital

in the short run we can talk about

factor productivity



In the long run all inputs become

variable e.g. the long run is the period in

which a firm can adjust all inputsto changed conditions

in the long run we can talk aboutreturns to scale (compare latter

with economies of scale, which

is a cost related concept)



Short-Run Changes in Production

Factor Productivity

Units of K

Employed Output Quantity (Q)

8 37 60 83 96 107 117 127 128

7 42 64 78 90 101 110 119 120

6 37 52 64 73 82 90 97 104

5 31 47 58 67 75 82 89 954 24 39 52 60 67 73 79 85

3 17 29 41 52 58 64 69 73

2 8 18 29 39 47 52 56 52

1 4 8 14 20 27 24 21 17

1 2 3 4 5 6 7 8

Units of L Employed

How much does the quantity of Q

change, when the quantity of L isincreased?



Long-Run Changes in Production

Returns to Scale

Units of K


8 37 60 83 96 107 117 127 128

7 42 64 78 90 101 110 119 120

6 37 52 64 73 82 90 97 104

5 31 47 58 67 75 82 89 954 24 39 52 60 67 73 79 85

3 17 29 41 52 58 64 69 73

2 8 18 29 39 47 52 56 52

1 4 8 14 20 27 24 21 17

1 2 3 4 5 6 7 8

Units of L Employed

How much does the quantity of Q change,when the quantity of both L and K is

increased?



Relationship Between Total, Average,and Marginal Product: Short-Run

Analysis

Total Product (TP) = total quantity of

output

Average Product (AP) = total product

per total input

Marginal Product (MP) = change in

quantity when one additional unit of

input used



The Marginal Product of Labor

The marginal product of labor is theincrease in output obtained by adding 1unit of labor but holding constant theinputs of all other factors

Marginal Product of L:

MPL= ∆ Q/∆ L (holding K constant)

= δ Q/δ L

Average Product of L:

APL= Q/L (holding K constant)



Short-Run Analysis of Total,Average, and Marginal Product

If MP > AP thenAP is rising

If MP < AP thenAP is falling

MP = AP when APis maximized

TP maximizedwhen MP = 0



Law of Diminishing Returns(Diminishing Marginal Product)

Holding all factors constant except one, thelaw of diminishing returns says that:

As additional units of a variable input arecombined with a fixed input, at some pointthe additional output (i.e., marginalproduct) starts to diminish e.g. trying to increase labor input without

also increasing capital will bringdiminishing returns

Nothing says when diminishing returns willstart to take effect, only that it will happenat some point

All inputs added to the production processare exactly the same in individual

productivity



Three Stages of Production in Short Run

AP,MP

X

Stage IStage II

Stage III

APX

MPX

Fixed input grosslyunderutilized;specialization andteamwork causeAP to increasewhen additional Xis used

Specialization andteamwork continue toresult in greateroutput whenadditional X is used;fixed input beingproperly utilized

Fixed input capacityis reached;additional X causesoutput to fall



Stage-1 Stage-2 Stage-3Qu

a

n

ti

t

y

Input-L

L1 L2 L3

TP

MP

AP

Increasing

Return

Decreasing

Return

NegativeReturn

Stage-2 is rational and Stage-1 and 3 are irrational



How to Determine the OptimalInput Usage

We can find the answer to this fromthe concept of derived demand

The firm must know how many units of output it could sell, the price of theproduct, and the monetary costs of employing various amounts of theinput L

Let us for now assume that the firm isoperating in a perfectly competitivemarket for its output and its input



Optimal Decision Rule:

A profit maximizing firm operating in

perfectly competitive output and input

markets will be using optimal amount

of an input at the point at which themonetary value of the input’s marginal

product is equal to the additional cost

of using that input (L)

- in other words, when MRP = MLC



Production in the Long-Run

All inputs are now considered to be

variable (both L and K in our case) How to determine the optimal

combination of inputs?

To illustrate this case we will use productionisoquants.

An isoquant is a curve showing all possible

combinations of inputs physically capable of

producing a given fixed level of output.

It is also known as Iso-Product Curve, Equal

Product Curve, Production Indifference Curve



Example 2 Production Table

Units of K


8 37 60 83 96 107 117 127 128

7 42 64 78 90 101 110 119 120

6 37 52 64 73 82 90 97 1045 31 47 58 67 75 82 89 95

4 24 39 52 60 67 73 79 85

3 17 29 41 52 58 64 69 73

2 8 18 29 39 47 52 56 52

1 4 8 14 20 27 24 21 171 2 3 4 5 6 7 8

Units of L

IsoquantUnits of K Employed



An Isoquant

Graph of Isoqua

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7X

Y



Substituting Inputs

There exists some degree of

substitutability between inputs.

Different degrees of substitution: (Different

Shapes)

Sugar

a) Perfect substitution b) Perfectcomplementarity

All otheringredients

Naturalflavoring

Q

Q

Capital

LaborL1 L2 L3 L4

K1

K2

K3

K4

Corn

syrup

c) Imperfect substitution

Linear

Isoquants Right Angle Convex



Substituting Inputs continued

In case the two inputs are

imperfectly substitutable, theoptimal combination of inputs

depends on the degree of

substitutability and on the relative

prices of the inputs



Substituting Inputs continued

The degree of imperfection insubstitutability is measured with marginalrate of technical substitution (MRTS):

MRTS = ∆ L/∆ K

(in this MRTS some of L is removed from

the production and substituted by K tomaintain the same level of output)



Law of Diminishing Marginal Rateof Technical Substitution:

Table 7.8 Input Combinatio

for Isoquant Q = 52

C ombination L K

A 6 2

B 4 3

C 3 4

D 2 6

E 2 8



Law of Diminishing Marginal Rate of Technical Substitution continued

0

1

2

3

4

5

6

7

2 3 4 6 8X

Y

∆ X = 2∆ Y = -1

∆ X = 1

∆ Y = -1

∆ X = 1

∆Y =- 2

A

BC

DE



MRTS = ∆ L/ ∆ K = - MP L /MP K

Units of K


8 37 60 83 96 107 117 127 128

7 42 64 78 90 101 110 119 120

6 37 52 64 73 82 90 97 1045 31 47 58 67 75 82 89 95

4 24 39 52 60 67 73 79 85

3 17 29 41 52 58 64 69 73

2 8 18 29 39 47 52 56 52

1 4 8 14 20 27 24 21 171 2 3 4 5 6 7 8

Units of L



M P L/ M P K in R e l a t i o n t o M R T S ( X f o r Y )

C o m b i n a t i o nQ L M P L K M P KM R T S ( L f o r K )M P L/ M

A 5 2 6 2B 5 2 4 1 3 3 6 , 5 2 2

C 5 2 3 1 1 4 1 1 1 1

D 5 2 2 6 , 5 6 1 3 1 / 2 1 / 2



The Production Isocost

The isocost function is the set of allcombinations of capital and labour that can

be purchased for a specified total cost.

C= rK + wL (r=Price of K w=price of L)

Rate of capital

Rate of

labour

16.7

13.3

10

15 20 25

C=Rs.50

C=Rs.40

C=Rs.30



Changes in the budget amount, C cause

the isocost line to shift in a parallel manner.

Changes in either the price of labour or

capital cause both the slope and one

intercept of the isocost function to change.

Rate of capital

Rate of labour

C=Rs.40

40/2

40/3

40/5

40/2

40/5 40/3



Optimal Decision Rule

Efficient production requires that theisoquant function be tangent to the

isocost funtion at this point;

MPL= MP

K

w r

Rate of L

Rate of k




There are three ways approaching to

optimal resource allocation

Maximize production for a

given rupee outlay on L and KMinimize the rupee outlay on L

and K to produce specified rate

of output

Produce the output rate thatmaximizes profit.

Constr

ained

optimization

proble

m




a

b

c

d

Rs.100

Rs.150

20

10

Rate of K

Rate of L



Optimal decision Rule

If the condition for efficient

production is not met, there is some

way to substitute one input for the

other that will result in an increase atno change in total cost.

Profit maximization requires that

inputs be hired until MRPK=r and

MRPL=w



Economies of Scale

Economies of ScaleA given rate of input of K and L

defines the scale of production.

Change in Scale

Proportionate changes in both inputs

Return to scale

The magnitude of the change in the

rate of output relative to the change inscale.

For eg. P.F. is Q= f(K,L) then

hQ= f( K, L)ƛ ƛh factor change in output when ƛ factor change in both variable



Sources of Economies of

Scale

Reasons for increase

in ROS

Technologies Specialization of

labour

Inventory economy

(increase or replacement)

Reasons for decrease

in ROS

Poor mgt. with thegrowth of the firm.

Transportation cost

Employing large no.

of local labour force.



Economies of Scope

It refers to per-unit cost reductions

that occur when a firm produces two

or more products instead of just one.

Firm can make use of its;Excess capacity

Unique skill or comparative

advantage in marketing to develop

complementary products.

For eg. :- P&G sells all kinds of cleaning

products as well some are complementary such

as laundry det., bleach, and fabric softeners.



Measure of economies of scale

S =TC(QA) + TC(QB) – TC(QA,QB)

TC(QA,QB)



Estimating production function

Ch-4 Production Theory

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