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MICROECONOMICS 1 – PRODUCTION THEORY

We continue from where we left off, examining some

concepts in production theory.

As we saw earlier, the Marginal Rate of Technical

Substitution (MRTS) measures the trade off between two

inputs in production with output constant.

1LECTURE MATERIAL ON MICROECONOMICS 1: PREPARED BY DR. EMMANUEL CODJOE


A formal derivation of the formula for the MRTS is given below.

Suppose our production function as generally specified is given by

Then if we consider the change in the use of factors 1 and 2 that

keeps output unchanged, then we have:


),( 21 xxfy

LECTURE MATERIAL ON MICROECONOMICS 1: PREPARED BY DR. EMMANUEL CODJOE 3


),(

),(

),(),(

,0),(),(

212

211

1

2),(

22121211

22121211

21 xxMP

xxMP

x

xMRTS

xxxMPxxxMP

xxxMPxxxMPy

xx


It is worth highlighting that the Marginal Rate of Technical

Substitution (MRTS) is also the slope of the isoquant.

That is,


211 ,

1

2

1

2 | xxyy MRTSx

x

dx

dx


Another closely related assumption about the nature of technology

embodied in the production process is that of Diminishing

Marginal Rate of Technical Substitution (MRTS).

That is, as we increase the amount of one factor, say x1, and adjust

the second factor, say x2, so as to stay on the same Isoquant, the

MRTSx1,x2declines.



It means that the slope of the MRTS must decrease in

absolute value as we move East and must increase as we

move North

Not to be confused with the law of diminishing marginal

product (law of diminishing marginal returns).


7

MRTSL,K is high; labour is scarce

so a little more labour frees up

a lot of capital

K

L

•

•

MRTSL,K is low; labour is

abundant so a little more

labour barely affects the

need for capital

LECTURE MATERIAL ON MICROECONOMICS 1: PREPARED BY DR. EMMANUEL CODJOE



Diminishing MRTS and diminishing marginal returns are

closely related but are not exactly the same.

Diminishing marginal returns is an assumption about how

the marginal product changes as we increase the amount of

one factor, holding the other factor fixed.



But diminishing MRTS is about how the ratio of the

marginal products – the slope of the isoquant – changes as we

increase the amount of one factor and reduce the amount of

the other factor so as to stay on the same isoquant.



Elasticity of Substitution: the MRTS despite its insights has

a serious defect, in that it is dependent on the units of

measurement of the factors.

Hence a better measure of the degree of factor substitution is

the elasticity of substitution.



Elasticity of Substitution: this is defined as the percentage

change in the capital-labour ratio, divided by the percentage

change in the MRTS.

Formally, the elasticity of substitution, can be

represented by the following expression:


LECTURE MATERIAL ON MICROECONOMICS 1: PREPARED BY DR. EMMANUEL CODJOE 12



Factor Intensity: the factor intensity of any process is measured

by the slope of the line through the origin representing the

particular process.

In other words, the factor intensity is the capital-labour ratio at

the particular point of interest in the production process.




A

B

100

30

50 200 L

K

Q=12


In the figure above, what is apparent regarding points A and B are that:

That is, the upper part of the isoquant is more capital-intensive than

the lower part, which includes more labour-intensive techniques



In general therefore, a production method that uses relatively

more labour than capital is labour-intensive, while a method

that uses relatively more capital than labour is capital-

intensive.



Returns to Scale: now suppose that instead of increasing one

input whilst the other is held unchanged, we increase both inputs

in the production process.

In other words, let’s scale the amount of all inputs up by some

constant factor, e.g., use three times as much of x1 and x2



Returns to Scale describes the relationship between inputs and output

when all factors of production vary. In other words, it describes the

output response to a proportionate increase of all inputs.

In general, if we scale all inputs by some amount, t, then three

possibilities can arise: constant returns to scale, increasing returns to

scale and decreasing returns to scale.



Returns to Scale are easily defined for homogeneous production

function. A production function is homogeneous of degree k if

where k is a constant and t is any positive real number.


),(),( 2121 xxfttxfxfk


Constant Returns to Scale: this arises if we use twice as much

of each input and we get twice as much output. Thus, in the case

of two inputs,


2� �1, �2 = �(2�1, 2�2) �� 1 , �2 = �(��1 , ��2)


Why might we expect this outcome? It should be possible for the firm

to replicate what it was doing before. Thus, if the firm has twice as

much of each input, it can just set up two plants side by side and

thereby get twice as much output.

But it is perfectly possible for a production technology to exhibit

constant returns to scale and diminishing marginal product to each

factor.



Increasing Returns to Scale: this arises when we scale up all

inputs by some factor, t, we get more than t times as much

output.

Mathematically, this is given by

for all t > 1


�� 1, �2 > �(��1 , ��2)


Decreasing Returns to Scale: this arises when we scale up all

inputs by some factor, t, we get less than t times as much output.

Mathematically, this is given by

for all t > 1


�� 1, �2 < �(��1 , ��2)


Returns to Scale: in general if both inputs are increased by the

factor t, and output is increased by the factor tk , returns to scale

are increasing if k > 1, constant if k = 1, and decreasing if k < 1.



Economies of Scale Vs. Returns to Scale

A production process is said to exhibit economies (constant

economies, diseconomies) of scale over a particular range of

output per unit of time if the long-run average production costs

fall (remains unchanged, increases) as output increases.



Economies of Scale Vs. Returns to Scale

The term returns to scale refers to the effect on output of a

proportionate change in the level of use of all inputs.

It is therefore important not to confuse the two concepts.


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