Barebones Outline of Neoclassical Growth Theory
revised 2006
Background to the classroom presentation and discussion.
© David M. Nowlan 2005
Unlike multiplier or export-base regional growth models, which are demand driven,
neoclassical growth models are supply driven. Markets are assumed to work in
neoclassical fashion so that production factors – capital and labour -- are kept more-orless fully employed; the source of growth in output therefore becomes the growth in the
factors themselves, including a somewhat elusive “technical change” component which
increases the productivity of some or all of the factors.
Typically, two factors of production are considered, physical capital denoted by the
symbol K , and labour denoted by L . A flow of labour input, say hours worked per year,
works together with some stock of capital to produce a flow of output Y according to
some functional relationship, Y  F ( K , L) for some given time period. Leaving the
technical change component aside for the moment, time subscripts can be put on the
variables to remind us that the inputs and output are in fact functions of time. We then
get
Yt  F ( K t , Lt )
Equation 1
The function F is quite general. This needs to be specialized in order to reflect the
neoclassical view that factors of production are paid their marginal products and total
payments to factors equals total output. So, in the neoclassical world
Yt
Y
K t  t Lt  Yt or MPK K t  MPL Lt  Yt
K t
Lt
Equation 2
In order for the function F to have the property shown in equation 2, it must exhibit
constant returns to scale, according to Euler’s Theorem. This means that if you scale up
each of the two factors of production by some constant proportion, say a , then output
will change by the same proportion. If F (aK t , aLt )  aYt , then F exhibits constant
returns to scale or CRS.
A CRS function can have a number of forms but the most commonly used form is called
a Cobb-Douglas function, which has the following specification:
Yt  K t L1t , where   1
Equation 3
You can check to see that if, for example, you double the input of both K and L , then
output will also double.
It turns out that the assumption of CRS, which is necessary within the neoclassical view
of the world, and a Cobb-Douglas functional form has very important implications for
growth theory. It entails the result that if one of the factors of production, say labour, is
held constant while the other grows in size, the marginal product of the growing factor
will gradually fall. More generally, if one factor grows relative to the other, its marginal
product will fall. Thus the ability of additional units of the faster growing factor to
increase regional or national output diminishes as the proportion of this factor
increases.
To see this, take the marginal product of capital derived from the Cobb-Douglas function,
equation 3. (I’ve dropped the time subscript in equation 4 to make it easier to read, but it
is implied.)
Y
L
 K  1 L1    
K
K
1
Equation 4
The equivalent equation for the marginal product of labour would be
Y
K
 1    
L
L

Equation 5
Notice first that the marginal products of both capital and labour are functions of the
capital:labour ratio only; the absolute magnitudes are not relevant. This is a property of
CRS functions. Notice further and importantly that, from equation 4, the marginal
product of capital will fall as the labour:capital falls (i.e., as capital becomes more
abundant relative to labour). From equation 5, we see that the marginal product of labour
similarly falls as labour becomes relatively more abundant.
Another way of looking at this is to note that with the Cobb-Douglas production function,
Y
output per worker, , which is often what we’re interested in as a measure of the well
L
1
being of a region, is a diminishing function of the amount of capital per capita. To see
this, take equation 3 and divide each side by the amount of labour:
Y
. But if it is assumed that the number of workers is a
N
L
Y
constant proportion of the total population, i.e.,
stays constant, then movements in
will be exactly
N
L
Y
reflected in movements in
. I will adopt that assumption and use growth in output per capita and
N
1
Usually we look at output per capita, say
growth in output per worker interchangeably.
2
Y K  L1  K 

 
L
L
L

Equation 6
Recalling that   1 , equation 6 shows us that a doubling of the amount of capital, for
example, with the amount of labour held constant, would result in less than a doubling of
output per worker. Growth in either factor alone is not enough to result in an equiproportionate growth in regional output. For example, the more you tried to increase
capital by raising the rate of saving in a region, the less each additional unit of capital
would contribute to a rise in output, as equation 4 demonstrated. More about this below.
Notice from equation 6 that, with a CRS production function, we can write output per
person as a function simply of capital per person; output per person depends only on the
capital:labour ratio. Using lower case letters to denote per person amounts, we have from
equation 6
y  k
Using equation 3, we can now look growth in this regional economy. Before doing this,
a few points are worth establishing. First, notice that, with factors paid their marginal
products, the share of each factor is equal to the exponent of that factor in the production
function, i.e., the share of output going to capital is  and the share going to labour is
1   . (Naturally the shares add up to one.)2
The rate at which a variable is changing at any point in time is given by the time
dY
derivative of the variable. For output, for example, this would be
(or, if you don’t
dt
Y
like derivatives you could write
; if this is the change for one time period, then t
t
would equal 1). The growth rate of the variable is this rate of change divided by the
dY 1
level of the variable. Again for output, the growth rate of output is given by
(or
dt Y
Y 1
).
t Y
2
To see this for capital, take the marginal product of capital, from equation 3:
 K  L1
Y
 K  1 L1   
K
 K

Y 
     . Multiply this marginal product by the number of units of
K

Y K
Y K
     .
capital and divide by total output to get capital’s share:
K Y
KY
3
For growth rates, I will use the symbol G . So GY denotes the rate of growth of
output, GK the rate of growth of capital, G L the rate of growth of the labour force,
G y the rate of growth of per capita output and so on.
Finally, notice that the rate of growth of a variable may also be shown as the time
dY 1 d ln Y

derivative of the log of the variable. So,
. (For proof see footnote.3)
dt Y
dt
Back to looking at growth in this economy. The easiest way to do this is by taking logs
of equation 3 and then taking time derivatives as discussed in the preceding paragraph.
ln Yt   ln K t  1   ln Lt
Equation 7
From this,
d ln Yt
d ln K t
d ln Lt
, or

 1   
dt
dt
dt
GY  GK  1   GL
3
Equation 8
To show this set Yt  Y0 e , where
gt
Y0 is some initial level of output, g is the growth rate of output and
Yt is output at time t . Take the log of each side: ln Y t  ln Y 0  gt ln e  ln Y 0  gt , since
ln e  1 .
d ln Yt d ln Y0
dt
Now take the time derivative:

 g  0  g.1  g . So the time derivative of the
dt
dt
dt
log is indeed the rate of growth.


1
A further note on the constant e used above. Recall that, by definition, e  lim 1   , as   .
 
Now, consider some annual rate of growth, i , like perhaps the annual rate of interest on a savings account.
t
If you start with an amount A0 in the account then it will become At  A0 1  i  after t years. Suppose,
however, that the interest rate is paid monthly and compounded. Then, after t years,
i 

At  A0 1  
 12 
12t
nt
i

. If the compounding is n times per year, we have At  A0 1   . Let
 n
nt
it
it


1  

 A0 1  
. As the number of
   


compounding times per year increases without limit, i.e. as n   and so    , this expression

i
1
i 1

 . Then we have At  A0 1    A0 1  
n 
 n
 
becomes At  A0 e , a short-hand way of describing continuous growth at i per cent per year.
it
4
In per capita terms, the growth relationship is this:
G Y  GY  G L  G K  1   G L  G L  G K  G L  Gk
Equation 9
L
Equation 8 says that the rate of growth of output in this economy is a weighted average
of the rates of growth of capital and of labour. Remember   1 ; this means that the rate
of output growth must always be between the rate of growth of capital and the rate
of growth of labour, unless capital and labour are both growing at the same rate, in
which case output will also grow at that rate.
To keep things simple at this point, suppose that the rate of growth of population, and
therefore of the labour force, in this region is given to us – it’s not something that we’ll
derive from within the model. Let this given growth rate be G L .
What about the growth of capital? If this is a region closed to outside capital flows, then
any growth in the capital stock must come from savings in the region. Suppose the
savings rate is a constant proportion of output, s . The annual addition to capital stock
will therefore be sYt for any year t . The growth rate of capital is therefore:
GK 
sY
K
Equation 10
Using equation 8 and these growth rates we can see how this regional economy will
evolve over time. If capital is growing faster than labour, i.e., if GK  GL , then output
must be growing more slowly than capital, since its growth rate is always between the
growth rates of the two factors of production, GY  GK . This means that the right-handside in equation 10 must be getting smaller over time, i.e., the rate of growth of capital is
diminishing. This will be true as long as GK  GL . Since G L is given and fixed, GK as
it declines must be converging on G L . Similarly, the rate of output growth must be
falling, and it too will continue to fall until all three growth rates have become equal, i.e.,
there will be a convergence to GY  GK  GL .
If we started from a situation where capital was growing more slowly than labour, then
output must be growing faster than capital. From equation 10, this means that the growth
of capital must be increasing. Once again it will continue to increase until all growth
rates have converged, again to the labour growth rate.
So, from anywhere we start in this model, the long term growth will always converge on
the growth rate of labour. This may be illustrated with a diagram, Figure 1 shown below.
5
In this diagram, growth rates of the variables are plotted against the capital:output ratio,
Y
. The growth rate of labour, G L , is a given constant. The growth rate of capital, GK , is
K
sY
given by
. Where G L and GK cross, both growth rates are equal and this must also be
K
Y
equal to the growth rate of output. Since
is constant at this point, this can be
K
considered the long-run equilibrium level of growth in this region, shown as E in the
diagram.
Suppose the economy is not initially at equilibrium but starts at a lower-than-equilibrium
ratio of output to capital, say at position 1 in Figure 1. At position 1, the labour force is
growing faster than the capital stock. Output also will be growing faster than capital
Y
(although slower than the labour force), so the
ratio will increase; the economy will
K
Y
shift to the right along the
axis until equilibrium is reached at E .
K
In the process of moving from the initial position 1 to equilibrium E , output per capita or
Y
Y
Y
per worker, , falls as
rises.4
will gradually stabilize as E is approached. In the
K
L
L
Y
long-run equilibrium,
remains constant.
L
If the economy initially was at 2 in Figure 1, with a higher output:capital ratio and a
Y
Y
lower output:labour ratio, then
would, over time, fall to the equilibrium while
K
L
increased.
The growth rates in output per capita and the movement in per capita output itself,
corresponding to the starting points 1 and 2, are shown in Figures 2 and 3. Suppose
initial positions 1 and 2 represent the current situation in two otherwise identical regions.
4
In this model,
Y
Y
and
always move in opposite directions. We can show this. Starting from the
K
L
Y L
 
production function, equation 3, we get
K K

Y K
L
   
L L
K
and inversely with

1

1

Y
  
  
 K 







1

1
L  Y  1
which gives
   . Using this, we get
K K

K
Y
 Y  1  K  1
 
   . So, moves positively with
L
Y
K
Y 
Y
.
K
6
In region 1 output per capita will start off higher than in region 2. But notice that the
higher output per capita in region 1 will be falling as the economy moves towards
equilibrium. In poorer region 2, output per capita will be rising. Ultimately output per
capita will reach the same long-run equilibrium level in each region. There will have
been a convergence to this same level of output per capita.5 If this model approximates
the real-world of economic regions, we should expect to see rich and poor regions,
measured in terms of output per capita, converging towards each other – poor regions
should be growing faster than rich regions. In the simplest case, the one we’ve been
working with so far, this convergence should be towards a uniform output per capita.
This is known as absolute convergence. We’ll discuss in class the evidence for and
against absolute convergence across countries and across subnational regions.
Sticking with our two regions, suppose that initially they are both in a growth equilibrium
like position E in Figure 1. This is repeated in Figure 4. Now suppose that one of the
regions, region 2 say, decides to try to improve its economic performance by
encouraging, one way or another, a higher savings rate s in the region. If it is successful,
s 1Y
s 2Y
the function showing the growth of capital in Figure 4 will shift from
to
. For
K
K
this economy, the growth equilibrium will shift from E 1 to E 2 . Output per capita will
initially rise but this growth will disappear as the new equilibrium is reached. At the new
Y
Y
equilibrium, the ratio
will be higher than initially, but the growth rate of
will be
L
L
just what it was at the beginning. (In this simple model without technical progress, the
Y
equilibrium growth rate in
is zero. This equilibrium growth rate will become positive
L
when technical change is introduced.)
Figure 5 shows the path of output per capita over time in these two regions. The two start
Y
off with the same
level. After region 2 succeeds in raising the savings rate, its output
L
per capita rises but then reaches a new level with no further growth. Both regions have
Y
either remained at or converged to, once again, a constant rate of growth in
(zero in
L
this case). Convergence to a constant rate of growth but at different levels of output
per capita is called conditional convergence. In this case, different savings rates are
the conditioning factor. Even in more sophisticated neoclassical models, this
convergence to common growth rates remains the dominant feature.
5
This result is a consequence of the fact that with neoclassical production function, economies with a
relatively low output per capita and correspondingly a relatively high output per unit of capital have a high
marginal product of capital. Not only is the growth rate of capital relatively high in a region like region 2
(see the
sY
function in Figure 1), but the marginal product of each additional unit of capital is high. In the
K
other, initially richer region, the marginal product of capital is low.
7
Figure 1
10%
sY
K
growth rates
8%
E
6%
GL
4%
2%
2
1
0%
Y/K
Figure 2
6%
hi Y/K, lo Y/L
4%
2
Y/L growth rate
2%
0%
-2%
-4%
1
lo Y/K, hi Y/L
-6%
time
Figure 3
10
(Y/L)
8
6
1
4
2
0
2
time
8
Figure 4
s 2Y
K
10%
growth rates
8%
s 1Y
K
E2
GL
6%
E1
4%
2%
0%
Y/K
Figure 5
10
(Y/L)
8
hi savings
6
lo savings
4
2
0
time
9
It’s time now to take technical change or productivity improvements into account.
Based on recent economic experience we expect the productivity of factor inputs to
improve over time. In many neoclassical models, this is captured by multiplying the
Cobb-Douglas production function by a term, At , that grows over time:
Yt  At K t L1t
Equation 11
In this formulation, technical change or productivity growth floats over the whole
productive process embodied in the Cobb-Douglas function. There’s no indication of the
origin of the technical change; in fact, the level and rate of change of At is given
exogenously and is independent of such things as the amount of investment in capital.
Because technical change is a multiplicative term in the production function, it can be
associated with either the capital or the labour term. Because our interest is in the growth
of output per capita, we associate it with the labour term in the following way:

1
t
Yt  At K t L
 1 
 K t  At1 Lt 



1
 K t Et 
1
Equation 12
The term in square brackets in equation 12 is called the “efficiency units” of labour
and, as you can see, these units are denoted by E t . As long as At is growing over time, the
efficiency units of labour will grow faster than labour itself. Thus,
 1 
GE  GL  
G A
1  
The same reasoning that took us to growth equation 8, yields from equation 11
GY  GA  GK  1   GL
Equation 136
This may be written as
G 

GY  G K  1   G L  A   G K  1   G E
1 

Equation 14
G A is sometimes referred to as the growth in “total factor productivity” which, from equation 13, is
GA  GY  GK  1   GL
6
10
The form of equation 14 is the same as that of equation 8, save that the growth in
efficiency units, G E , has replaced the growth in labour units, G L . Now, equilibrium
growth occurs where K and E are growing at the same rate, which will also be the
equilibrium rate of growth of output. This time, however, this equilibrium growth rate of
output will exceed the growth rate of labour, G L , as long as GA  0 . Output per capita
will therefore be growing in equilibrium as a consequence of introducing technical
G
change. We have, in equilibrium, GY  GE  GL  A . So the growth rate of output
1
GA
capita, which is GY  GL , is equal to
.
1
The new situation, with technical change, is shown in Figure 6, which is similar to Figure
1. Equilibrium now occurs where the growth of capital equals the growth of efficiency
units of labour.
If two regions are alike in all relevant respects – the same savings rate, the same
population growth and the same exposure to technical change, then the equilibrium of
each will be at E in Figure 6. Even if the regions start at very different output: capital
and output:labour ratios, say at positions 1 and 2 respectively in Figure 6, income per
capita in each region will converge to a common level and a common growth rate. This
again is an example of absolute convergence and is shown in Figure 7. A region starting
off with a lower level of per capita income will grow faster than one with a higher level.
They both converge to the same level and same growth rate.
(Notice that in Figure 7, the vertical axis is the log of income per worker. A constant rate
of growth therefore appears as a straight line.)
Introduce once again a higher savings rate in one of the two otherwise identical regions.
sY
Imagine shifting in Figure 6 the
line for one of the regions, as we did in Figure 4.
K
Y
Y
The region with the higher savings rate will end up with a lower
ratio and higher
K
E
Y
and
ratios, but the growth rate of output and of output per capita will, in equilibrium,
L
be the same in both regions. This is again an example of conditional convergence. The
higher savings rate has introduced a level effect, i.e., the level of output per capita is
higher in the higher savings region, but not a growth effect. Figure 8 illustrates
conditional convergence with a level effect only.
We saw in footnote 2 that the marginal product or return to capital is a direct function of
Y
the output:capital ratio, MPK   . This leads to the insight that the more open are
K
regions, open that is to capital and labour flows, the more likely it is that absolute rather
than conditional convergence will occur. Consider a region that has somehow achieved a
11
Figure 6
GK 
10%
E
growth rates
8%
sY
K
GE
6%
4%
GL
2%
1
2
0%
Y/K
Figure 7
10
ln(Y/L)
8
starting at position 1
6
4
2
starting at position 2
0
time
Figure 8
hi savings
10
ln(Y/L)
8
lo savings
6
4
2
0
time
12
higher rate of savings. As we have seen, this leads to a level effect in per capita income.
Y
It also leads to a level effect in capital intensity, the
ratio: this ratio will fall as the
K
capital intensity rises, as you can see by looking at Figure 4 or considering footnote 4.
This in turn means that the marginal product or return to capital in this region will fall
relative to the region that didn’t raise its savings rate. Savers will now be attracted to the
region with the higher return to capital – this is where they will want to invest – so some
of the savings in the higher-savings region will be drawn into the other region. With
complete openness to capital flows and other relevant circumstances identical, this flow
of savings or capital from one region to the other will continue until the returns to capital
sY
are the same in the two regions. The adjusted
function – the rate of growth of capital
K
– will then be the same in each region and the openness will have the result that the level
difference in output per capita between the regions will have disappeared. Convergence
will be absolute, not conditional, although because of the higher savings rate in one
region, both regions will have a somewhat higher level of output per capita. Long-run
growth rates will not have changed.
Neoclassical growth theory thus predicts that output per capita of regional or
national economies will converge over time. We should expect to see richer
countries growing more slowly than poorer countries. While this convergence may
be conditional on some factors that vary among regions, the more open the
economies the more likely it is that convergence will be absolute. This theorizing is
one of the principal bases for believing, in the European Union, that reduced trade
and factor-movement barriers will lead to regional economic convergence.
We will see in class that empirical support for absolute convergence among nations or
regions is weak, but there is more support for the existence of conditional convergence.
The most interesting recent writing on neoclassical growth has focused on the question of
what factors exactly – higher levels of schooling, democratic government, rule of law,
inflation rate and so on – are the conditioning factors that can produce level effects in
output per capita. Notice that many of these factors are not transportable from one region
to another through the open-economy mechanisms of capital and labour flows, so open
economies alone may not be enough to bring about absolute convergence. We will
explore some of these results in class.
All neoclassical models predict convergence in growth rates. In spite of the considerable
challenge to this position from models of economic geography and the new growth
theory, the introduction of level effects and conditioning variables into the basic
neoclassical model has allowed us to understand some of the factors that help a region
become richer, or that might keep it poorer.
13