Lecture Notes for Topic 4: Economic Growth
James R. Maloy, Department of Economics, Spring 2020
Readings: Froyen, Chapter 20 (8th Ed. Ch. 5) has a basic form of
the Solow model. These lecture notes present a simplified version
of a very detailed presentation of the Solow model from David
Romer’s Advanced Macroeconomics postgraduate textbook.
The purpose of this section is to understand the determinants of long-run
economic growth. Economic growth is the change in output (GDP) over
time. Long-run growth theory is concerned with what is typically called
trend or potential GDP, not short-run business cycles which are simply
fluctuations around this trend. Long-run growth is due to changes in supply
factors which affect the production abilities of societies (recall the vertical
long-run AS from the classical model–most growth theories are founded in
classical theory). The focus in long-run growth is usually per capita; it is
true that more population will allow more total GDP, but economic growth
theorists are typically more interested in the factors that affect output per
capita, which is known as labour productivity.
There are many important questions in growth theory, which typically
centre around two key ways of looking at growth: inter-temporally or cross-
sectionally. The first is to look at a particular society over a period of
time–why does a particular nation have more per capita output today than
in, say, 1872? The second approach asks why GDP (and GDP growth rates)
differ amongst nations at the same time–why is the US relatively affluent
while India is relatively poor? And furthermore, will India be able to catch
up in the future? The basic way to start answering these types of questions is
determine the factors which affect economic growth. Many of the much older
theories of economic growth focused on variables such as savings and capital
formation as the driving factors. However, the Solow Model, developed
in the 1950s and the original model behind much modern growth theory,
actually argues that such factors are not the most important things.
2 Assumptions and Variables in the Solow Model
Let us begin by identifying some variables:
Y = output
K = capital stock
N = labor
A = effectiveness of labour. This is basically a variable that encom-
passes all factors that determine how effective labor is, such as knowledge,
technology, etc. Often this is just simplified to technology only.
AN = effective unit of labor
y = YAN : This is defined as output per unit of effective labor.
k = KAN : This is a similar measure of capital per unit of effective labor.
Note the difference between Y and y, and K and k. They are distinct
items; do not get them confused. Now that we have some basic ideas, let us
define an aggregate production function:
Y = F (K,AN) (1)
Output is a function of the capital stock and the amount of effective
labour. More specifically, we can define a Cobb-Douglas production function
Y = Kα(AN)1−α (2)
Note that this production function has constant returns to scale (the
exponents sum to 1). This greatly simplifies the model. However, the model
relies not on the aggregate production function Y , but on the intensive form
production function, y. Recalling our definition of y from above, dividing
the above production function by AN yields:
y = Y
AN = Kα(AN)1−α
ANα = (
AN )α = kα (3)
Therefore, the intensive form production function is:
y = kα (4)
This production function gives output per unit of effective labour. Graph-
ically, it looks like the familiar production function from topic 2, only note
that the axes are different; the vertical axis is y and the horizontal axis is k.
To make this a model of growth, we must introduce time into the model.
Specifically, all of our variables in the original production function (for total
output Y ) are functions of time (t):
Y = [K(t)]α[A(t)N(t)]1−α (5)
Therefore, output at time t depends on that period’s capital stock, labor
force, and the level of technology and other factors that determine labour
effectiveness. In order to finish building the model, Solow makes some as-
sumptions about the growth rates of these items. The growth rate is calcu-
lated by taking a time derivative, i.e. a derivative of each item with respect
to time. A time derivative is symbolised by placing a dot over the variable.
Much of this math involved in this requires some knowledge of differential
equations, which if you have not taken Calc II you will not know how to do,
so don’t get bogged down in worrying about where these equations come
from as it really isn’t necessary to understand the model.
1. Assumption 1: The labor force N grows at a constant, exogenously
given rate n.
2. Assumption 2: The level of effectiveness of labor A grows at a constant,
exogenously given rate g.
3. Assumption 3: Investment in each period is some fraction of output.
We can assume that investment is equal to the fraction of output that
is saved, e.g. all savings are channelled to investment (like in the
classical loanable funds theory with a balanced government budget).
This can be expressed as I(t) = sY (t). Therefore, the amount invested
each period is determined by the savings rate s, where 0 ≤ s ≤ 1. The
savings rate s is also assumed to be constant and given exogenously.
4. Assumption 4: Finally, we must indicate how the (total) capital stock
K changes over time, i.e. the value of K̇. It is assumed that the rate
of change of the capital stock is a function of the level of investment
(from assumption three) and the depreciation of the existing capital
stock. This can be given by K̇(t) = sY (t) − δK(t)
Therefore, the rate of change of the capital stock over time is the
level of investment minus the amount of depreciation, δ. Note that
0 ≤ δ ≤ 1. It is also taken exogenously.
3 The Solow Equation and the Solow Diagram
We are now ready to formulate the “Solow equation”. The key to the Solow
model lies with the time derivative of k (capital per unit of effective labor).
Specifically, this is given by
k̇(t) = ˙
A(t)N(t) ) (6)
This is solved by differentiating and plugging in values for the variables.
The mechanics are given by Romer but rather than working it out, which
makes no one happy except for the one person in this course who actually
enjoys the chain rule, I will save us a headache and simply assert that this
will yield the key Solow equation. Using the Cobb-Douglas intensive form
production function from equation 4 above, this is given by:
k̇(t) = skα − (n+ g + δ)k (7)
This equation has two components. The first component indicates that
savings increases the rate of change of capital per unit of effective labor over
time. Recalling assumption three above, this is because savings is channelled
into investment in new capital goods. The second component (which is
subtracted) indicates that the growth of labor n, labour effectiveness g, and
depreciation δ (see assumptions 1, 2 and 4 respectively) all decrease the rate
of change of capital per unit of effective labour. Showing each of these parts
of the equation separately on a graph (the Solow diagram) will help this to
The Solow diagram shows capital per unit of effective labor k on the
horizontal axis and investment per unit of effective labor on the vertical
axis (which is a fraction of output per unit of effective labor y, as noted
above in assumption 3). Recalling our production function, we note that
the first term in the Solow equation is just a fraction s of the production
function. Therefore, it is sloped just like the production function. The
second component is a linear function, since we assumed that n, g and δ
are constant. Note that the two lines intersect. We shall call the value of k
where the curves intersect k∗.
It is now necessary to explain these curves in more detail. The curved
line, skα, is the actual level of investment. Since we assumed that all savings
are chanelled into investment, the fraction s of output that is saved must
be the level of investment by definition. The straight line, given by (n +
g+ δ)k is what is called break-even investment. It is the level of investment
needed simply to maintain k at its current level. What does that mean?
Well, suppose initially that there are 5 units of capital K and 5 units of
effective labour AN . Therefore, k is equal to 1. Now suppose that you have
population growth n. Specifically, suppose that now AN = 6 due to this
increase in N . In order to maintain k = 1, you must invest in one more
unit of K to make K also equal to 6. This is why it is called break-even
investment. It is the amount of investment that is needed to maintain k at
its current level. Looking back at the Solow diagram, note that at values
of k below k∗, actual investment exceeds break even investment. Therefore,
the amount of capital per unit of effective labour is growing, e.g. k is getting
bigger. Looking back at the Solow equation, note that the first term is bigger
than the second term so the rate of change of k is positive. If k is greater
than k∗, then actual investment is below break even investment, and k is
getting smaller; from the Solow equation, the second term is larger then the
first term and the rate of change of k is negative. The model thus implies
that k converges to k∗, and k will be constant at that point. This is what
is known as the steady state or balanced growth path. The actual value of
k∗ can be calculated by setting the Solow equation equal to zero (since k
is constant at the steady state so its rate of change is zero) and solving for
k. Note that just because the economy reaches a steady state in which k
becomes constant does not mean the economy as a whole is not growing;
it just means that, in the steady state, all of the variables are growing at
a constant rate. The implication is that regardless of its initial starting
position, the economy will move to a long-run steady state, with constant
growth. We will now turn to a description of the steady state, and also
discuss how changes in the values of savings and other variables will cause
the economy to converge to a new steady state.
It was mentioned above that all variables grow at a constant rate in the
steady state. Some simple calculations can show that these growth rates
Variable Growth Rate
K n+ g
AN n+ g
Y n+ g
The most interesting conclusion is that output Y is growing at a constant
rate n + g. This indicates that in the steady state, the growth rate of
output depends on the growth rate of population and labour effectiveness.
Furthermore, the growth rate of output per worker Y/N depends only on
the growth rate of labour effectiveness A. This may come as a surprise; it
is often thought that the level of savings determines the growth rate of the
economy. The Solow model indicates that savings do not affect the growth
rate of output in the steady state. What we will see is that saving levels do
effect the level of output, but not the rate of growth of output in the steady
Changes in savings will only affect growth rates in the short run transi-
tion period. If output was growing at 5% before the increase in savings rates,
it will be growing at 5% once the new steady state is reached. This does not
mean that savings does not affect the economy—quite the contrary. During
the transition period, variables will be affected. This temporary change has
permanent effects on the level of variables. To think of an example, suppose
that every year you get a 10% raise on the previous year’s pay, which we say
is $100 in 2015. Now suppose that in 2016 you get a special bonus that dou-
bles your income to $200. The next year, you go back to getting 10% raises,
but this is now 10% of a much bigger value than it would be otherwise—it
is 10% of $200 instead of $110! Therefore, although after the transition
period the growth rate is back to 5%, you are at a higher level of income
thanks to the bonus. A similar type of thing is occurring with savings in the
Solow model. The Solow model indicates that, once the new steady state is
achieved, output will be growing at the same rate as before, say 5%, but the
change in savings will affect the level of output, and an increase in savings
means that we are now taking 5% of a larger level of output than we would
have had if savings had not increased. A few simple graphs can capture the
essence of a change in the savings rate. Suppose initially the economy is in
the steady state, and at time t0 there is a permanent increase in the savings
rate. This can be shown as:
We know from the Solow diagram that capital per unit of effective labour,
k, increases to the new steady state. Once the new k∗ is attained, the rate
of change of k is again zero.
The effects of the increase in savings on the natural logarithm of Y/N
can now be seen. Output per unit of labour climbs until the new steady
state is reached, when it becomes parallel to the original growth path, once
again growing at a rate of g.
4 Conclusions of the Solow Model
We have spent considerable time deriving and working with the Solow model,
but have yet to draw any fundamental conclusions. By going a bit deeper
than we have here, it can be shown that the Solow model indicates two
main sources of differences in output per worker Y/N over time (or across
nations). Differences in capital per worker K/N and differences in the ef-
fectiveness of labour A will affect the value of output per worker Y/N .
Furthermore, the model indicates that only growth in the effectiveness of
labour can cause a permanent change in growth rates. Before Solow, many
theorists had suggested that differences in capital stocks per worker were the
reason why some nations are rich and others poor; the Solow model indicates
that different capital stocks alone can not account for differences between
nations, or for that matter differences in a particular nation over time. The
key seems to rest with differences in the effectiveness of labour. However,
the Solow model assumes that the value of A and the growth rate of A are
exogenous and constant; therefore, it takes as given the very thing which
causes economic growth! Furthermore, Solow just treats A as a catch-all
phrase that incorporates everything except capital. A can include knowl-
edge, technology, human capital, property rights, attitudes towards work, or
anything that can determine labour effectiveness; often it is just simplified
to technology (Froyen does this) although Solow was not that specific. This
is a key weakness in the model; the model indicates that growth depends
on A but does not give any indication of what A is or how it is determined,
which has led to later theories that expand on this concept.
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