ECN115
Lecture 6
G. Renshaw’s ch. 17
Returns to scale and
homogeneous functions;
partial elasticities; growth
accounting; logarithmic scales
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Returns to scale
Assuming that both 𝐾 and 𝐿 inputs double production will be
moved to point 𝑁 where the output level is now 𝑄1 .
The crucial question is the relationship between the new level of
output (𝑄1 ) and the initial level (𝑄0 ).
There are three possibilities:
(1) If 𝑄1 = 2𝑄0 , then output has exactly doubled. Thus output has
increased in the same proportion as the increase in inputs. This is
the case of constant returns to scale.
(2) If 𝑄1 > 2𝑄0 , then output has more than doubled. Thus output
has increased in a greater proportion than the increase in inputs.
This is the case of increasing returns to scale (also known as
economies of scale).
(3) If 𝑄1 < 2𝑄0 , then output has less than doubled. Thus output
has increased in a smaller proportion than the increase in inputs.
This is the case of decreasing returns to scale (also known as
diseconomies of scale).
Returns to scale are variable, depending on the initial position,
but also possibly on the value of 𝜆.
This is because the isoquants in the figure to the left have
very different shapes from each other.
Another possibility is that there are constant returns to scale
when output is small (the initial position, J, is close to the
origin) but strongly increasing returns to scale when output is
large (J is far from the origin).
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Returns to scale and homogeneity of 𝑄 = 𝑓 𝐾, 𝐿 when 𝜆 = 2 (doubling 𝐾 & 𝐿)
Effect on Q when K and L are
both doubled (from any
initial position)
Economist’s view (assuming
Q = f(K, L) is a production function)
Mathematician’s view (assuming
Q = f(K, L) is a any function)
Q is exactly doubled
Function has uniform constant
returns to scale
Function has uniform increasing
returns to scale
Function has uniform decreasing
returns to scale
Function is homogeneous of
degree 1
Function is homogeneous of
degree greater than 1
Function is homogeneous of
degree less than 1
Q is more than doubled
Q is less than doubled
Testing for homogeneity
The procedure for testing for homogeneity is as follows. Given any function 𝑧 = 𝑓(𝑥, 𝑦):
Step 1. We suppose that we are initially at some point 𝐽 of a plain, where 𝑥 = 𝑥0 , 𝑦 = 𝑦0 and
consequently the value of 𝑧 is given by 𝑧0 = 𝑓(𝑥0 , 𝑦0 )
Step 2. We increase x and y by the factor 𝜆 their new values are therefore 𝑥1 = 𝜆𝑥0 and 𝑦1 = 𝜆𝑦0 .
The new value the value of 𝑧 is given by 𝑧1 = 𝑓 𝑥1 , 𝑦1 = 𝑓(𝜆𝑥0 , 𝜆𝑦0 )
Step 3. If we can show from the equation 𝑧1 = 𝑓 𝜆𝑥0 , 𝜆𝑦0 that 𝑧1 = 𝜆𝑟 𝑧0 , then the function is
said to be homogeneous of degree 𝑟. If we cannot, the function is not homogeneous
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Properties of homogeneous functions
Property 1: the 'parallel tangents' property
4
Properties of homogeneous functions
Property 1: the 'parallel tangents' property
RULE 17.1
The relationship between returns to scale and long-run average and marginal cost,
given constant input prices
If the production function is homogeneous of degree 1 (uniform constant returns to scale),
then average cost is the same at every level of output. The average cost curve, as a function of
output, is therefore a horizontal straight line. Consequently 𝑀𝐶 = 𝐴𝐶 = 𝑎 constant, at every
output level (like the top right figure in the previous slide).
If the production function is homogeneous of degree less than 1 (uniform decreasing
returns to scale), average cost increases with output. The average cost curve is positively
sloped, and marginal cost must be greater than average cost at any output
(like the bottom left figure in the previous slide).
If the production function is homogeneous of degree greater than 1 (uniform increasing
returns to scale), average cost decreases with output. The average cost curve is negatively
sloped, and marginal cost must be less than average cost at any output
(like the bottom right figure in the previous slide).
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Properties of homogeneous functions
Property 2: the relationship between marginal and average products
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Properties of homogeneous functions
Property 2: the relationship between marginal and average products
7
Properties of homogeneous functions
Property 3: Euler's theorem and the adding-up problem
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Partial elasticities
for any function 𝑦 = 𝑓(𝑥)
arc elasticity
is the difference quotient
point elasticity
is the derivative of the function
both elasticities measure the rate of proportionate change of y. This contrasts
with the difference quotient and the derivative , both of which measure the rate of absolute
change. The difference between the two elasticity concepts is that the arc elasticity measures the
average rate of proportionate change following a discrete change in x, while the point elasticity
measures the rate of proportionate change at a point on the function.
These concepts are easily extended to a function with two or more independent variables z = 𝒇(𝒙, 𝒚)
arc partial elasticity of z with respect to x
point partial elasticity of z with respect to x
Partial elasticities of demand
Suppose there are two goods, 𝑎 and 𝑏, and 𝑌 = consumers’ income. Let 𝑞𝑎 denote the total
quantity of good a demanded by all consumers, and 𝑝𝑎 and 𝑝𝑏 denote the prices of goods 𝑎 and
𝑏. Then the combined demands of all consumers, known as the market demand function for
good 𝑎, might be of the form
𝑞𝑎 = 𝑓(𝑝𝑎, 𝑝𝑏, 𝑌)
The proportionate differential of a function
total differential of a function 𝑧 = 𝑓(𝑥, 𝑦) is given by
(17.3)
‘If there occurs a small proportionate change in 𝑥, and at the same time a small proportionate
change in 𝑦, what is the resulting proportionate change in 𝑧?’
The proportionate differential of a function
𝑑𝑧
On the left-hand side we have , the proportionate change in 𝑧.
𝑧
On the right-hand side, the terms in brackets are the partial elasticities of the function.
𝑑𝑥
The partial elasticity with respect to 𝑥 is multiplied by , the proportionate change in 𝑥.
𝑥
Similarly, the partial elasticity with respect to 𝑦 is multiplied by
𝑑𝑦
,
𝑦
the proportionate change in 𝑦
So, in words, rule 17.3 is answering our question above as follows:
𝑑𝑧
‘The resulting proportionate change in 𝑧, , is given by the sum of the proportionate changes in 𝑥 and 𝑦, each
𝑧
multiplied by its partial elasticity.’
In effect, the partial elasticities act as weights on the proportionate changes in 𝑥 and 𝑦.
𝑥 𝑑𝑧
If for example
is large, then this acts as a heavy weight on so that the latter has a large effect on 𝑧.
𝑧 𝑑𝑥
Growth accounting
In analysing the growth of total output in an economy over time, the standard neoclassical
approach is to assume that there exists an aggregate production function for the whole
economy of the form 𝑄 = 𝐹(𝐾, 𝐿), where 𝑄 = total output of goods and services (GDP) and
𝐾 and 𝐿 are the total inputs of capital and labour.
the proportionate differential formula (rule 17.3) to this
aggregate production function gives
if we then expected the supply of capital to grow
(due to investment) by, say, 20% over the next 10 years,
and the supply of labour to grow by, say, 10% over
the same period, then we get
assume that the aggregate production function is of
Cobb–Douglas form with constant returns to scale: that is
then we know from rule 17.4 that the partial elasticities are
Growth accounting
then we know from rule 17.4 that the partial elasticities are
One way of estimating 𝛼 is to use the marginal productivity
conditions (see ch.16, lecture 03.7). The marginal productivity
conditions state that with perfect competition and profit maximization,
we will have
If we substitute the marginal productivity conditions into the
Partial elasticities, the partial elasticities become
(17.7)
Here 𝐾 × 𝑟 is the income received by owners of capital and 𝑄 × 𝑝 is aggregate income (GDP),
𝐾𝑟
So
is the share of aggregate income received by owners of capital, which we can measure
𝑄𝑝
by looking at the share of profits in GDP.
Similarly 𝐿 × 𝑤 is wage income, so is the share of wage income in GDP.
(These are known as factor shares because 𝐿 and 𝐾 are the factors of production.)
i. then equation (17.7) tells us that we can use the shares of profits and wages in GDP as
estimates of the partial elasticities 𝛼 and 1 − 𝛼 respectively.
ii. the methodology gives us a tool for explaining past growth of output by reference to the past
growth of inputs.
iii. the methodology helps also to predict the effects of an increase or decrease in the growth rate of inputs
Logs, logs, logs
Given any function, if we transform the function by taking natural logs, and then find the
differential of the resulting function, this is equivalent to finding the
proportionate differential of the original function.
If you are struggling to grasp all of these results involving logs, hold tight to the basic fact
𝑑𝑦
that dln 𝑦 = for any variable 𝑦, provided 𝑑𝑦 is sufficiently small.
𝑦
In other words, for small changes, the absolute change in the log of a variable equals the
proportionate change in the variable itself.
Elasticity and logs
In ch. 13, lecture 05 we have seen that any function 𝑦 = 𝑓(𝑥)
Proportionate & absolute
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Partial elasticities and logarithmic scales
The proportionate differential and logs
Recall from rule 17.3 above that the proportionate differential
of the function 𝑧 = 𝑓(𝑥, 𝑦) is
This formula, derived from the total differential, gives the proportionate differential (that is,
as a function of the proportionate changes in 𝑥 and 𝑦, (
𝑑𝑥
𝑥
and
𝑑𝑦
𝑦
𝑑𝑧
𝑧
)
), and the partial elasticities.
It is used frequently in economics because we are often focusing on proportionate rather than
absolute changes.
In lecture 05, ch.13 we mentioned that if 𝑦 is any variable that increases by a small
Δ𝑦
amount, Δ𝑦, then
approximately equals Δ(ln 𝑦). That is, the proportionate change in 𝑦 is
𝑦
approximately equal to the absolute change in the natural log of 𝑦.
We can write this relationship as
The proportionate differential and logs
Rule 17.7 is true of any variable that changes for whatever reason.
Therefore for variables 𝑧, 𝑥, and 𝑦 we have
Log linearity with several variables
Any function of the Cobb–Douglas form 𝑧 = 𝑥 𝛼 𝑦 𝛽 , where 𝛼 and 𝛽 are constants,
has this property.
This log linear property continues to hold as the number of independent variables increases: for
example,𝑧 = 𝑥 𝛼 𝑦 𝛽 𝜈 𝛿 .
(Of course, most functions are not log linear. The assumption that some functional relationship is
log linear is merely a convenient simplification widely used in economics.)
This is a log linear function in three dimensions. If we constructed its three-dimensional graph,
with ln𝑄, ln𝐾, and ln𝐿 on the axes, the resulting surface would be a plane, with an intercept
of ln𝐴 on the ln𝑄-axis and slopes in the ln𝐾 and ln𝐿 directions of 𝛼 and 𝛽 respectively.
 Study:
from G. Renshaw’s ch. 17
 Attempt:
all relevant progress exercises
 for next week’s class/seminar prepare the so-called tutorial
exercises
 for next week’s teamwork prepare & upload the so-called
project exercises
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