Page 2 of 41THE MODEL SETUP AND QUESTIONSG.docx

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THE MODEL SETUP AND QUESTIONS
GDP (the demand side of the economy) is given simply by
our standard expenditure equation:
Y = C + I + G +NX
For these notes we make the simplifying assumption that
there is no government or exchange of goods and
services with the rest of the world. Hence, G = NX = 0 and
GDP (again, the demand side of the economy) is given
simply by:
Y = C + I.
You might be asked to think about what happens if there
is government and exchange with the rest of the world at
some point though. So you have to fully understand the
model to be able to tweak it, in case and answer those
questions.
We’ll look at an economy with given “structural
characteristics”:
A given production function ==> the Cobb Douglas
production function that we have studied already.
This represents the supply side of the economy.

A given exogenous savings rate for the economy: s
A given population growth rate: n
A given depreciation rate of capital: d
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With this info we want to analyze the economy long run
behavior…that’s what growth is all about. We want to try
to understand the evolution of GDP and other
macroeconomic variable with a long time horizon
perspective.
In particular, we want to analyze changes in the economy
over time:
We have seen so far that to affect productivity we need
to understand physical capital and investment so:
– How do these structural characteristics interact
to determine the investment level, and the
evolution of the capital stock?
– How does the evolution of the capital stock
interact with population in determining the
change in production?
– We’ll discuss how these factors determine the
behavior of the economy period after period,
and the implication of this for its long run
evolution.
What are the level of physical capital, output,
investment and consumption in the long run for
a specific economy?

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THE EQUATIONS OF THE MODEL
We have 5 basic ingredients (equations) in the Solow
model (yes, you need to memorize those and be able to
work the math out). Thankfully, we have seen 4 of these
5 equations previously at some point during this course
so it is just a matter of putting them together, and
understanding how they interact:
1) The production function: We have seen this equation
concerning the production function already in the slides
for chapter 12. For these notes we will use the Cobb
Douglas production function which, again, you have seen
in details. It has the constant returns to scale property.
Formally:
A is the TFP (or technology).
is physical capital at period t
is labor at period t
0 < < 1 is called the capital share you should know
this already.
1 is called the labor share you should know this
already.
Only 2 factors of productions (K, L) are analyzed jointly
with technology (A) here. This is for simplicity. It is
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possible to make the model more complicated and

consider more factor of productions such as human
capital, knowledge capital, organization capital. At this
stage, it’s also easier to think of the economy as
producing a single homogenous good (think of it as the
basket for all goods), and using just one type of labor (i.e.
there is no difference between skilled and unskilled
labor). But it is possible to modify all these assumptions.
2) At any point in time total national savings are just a
constant fraction of total national (disposable) income.
is national savings at period t
s is the national saving rate and it is expressed as a
percentage.
We have seen this equation and the national saving rate
already in the class notes for chapter 13 (the math of
saving). Note that here disposable income is equal to
total income because there are no taxes or transfers. This
can be easily changed by introducing those taxes and
transfers and government spending.
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3) Investment is equal to savings (aka the financial
market)
Since we are in a closed economy (without trade and
relationships with the rest of the world), everything that
is not consumed must be invested, so investment has to
be equal to savings.
We have seen this equation in chapter 13 already and in
the class notes concerning math for saving. This is
basically the equation representing the financial
system.

4) The expression for capital accumulation
is the future amount of capital
is the depreciation rate and it is expressed as a
percentage.
is the current amount of capital
Hence, this expression simply says that future capital
stock ( ) is equal to investment in new capital
plus current capital ( ) minus the capital that breaks
down and needs to be replaced ( )
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(machines/structures that break down or become
obsolete).
We have seen the equation above in class when we
explained investment.
5) Population/Labor force. To keep things simple, we
assume that the whole population of the country works
labor force is equal to the total population. There is
no unemployment or people outside the labor force.
Again, it is possible to modify this assumption at a cost of
mathematical complexity.
This is the only equation of the Solow model that you
have not seen already. But it is very simple to understand
it.
is the population growth rate and it is expressed as a
percentage. You can think of this as the net difference

between the birth rate and the death rate in the
economy.
So, the rate of growth of the labor force will be equal to
the rate of growth of the population.
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SOLVING THE MODEL: FROM FIVE EQUATIONS TO TWO
We can reduce the 5 equations of the Solow model just
down to 2 by doing some substitutions, here are the
precise steps:
First plug 2) into 3):
Now plug 6) into 4):
Finally, use 1) into 7):
This equation 8) together with equation 5) can be used to
fully characterize the Solow model. Those equations
describe how the two factors of production evolve over
time. Before we proceed further, let me try to convince
you numerically why the above steps are important in
practice.
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• A simple numerical example using the Solow model:
Suppose that periods are years, and that an
economy is characterized in the initial year by the
following “parameters”:

Y = K0.5L0.5; the initial level of capital and labor are
K1 = 10 and L1 = 5
s = 0.2 (20%), d = 0.03 (3%), n = 0.02 (2%)
Year 1: plug K1 and L1 into the production function
to obtain
Y1 = 10
0.550.5 = 7.1
Year 2:
Use the accumulation of capital equation and the
population equation to obtain the new capital
stock and new labor force
K2 = sY1 + (1 d)*K1
= 0.2x7.1 +(1 0.03)x10 = 1.41 + 9.7 = 11.11
L2 = (1+n)*L1 = 1.02x5 = 5. 1
Use these data to reiterate the process.
Y2 = 11.11
0.55.10.5 = 7.53
Growth rate of GDP between years 2 and 1 =
(7.53 7.1)/7.1 = 0.43/7.1 = 0.065 (6.5%)
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SOLVING THE MODEL: EXPRESSING THE MODEL IN PER
CAPITA TERMS
What do we do now? Well, to begin with we would like
to rewrite the model in per capita or per workers terms.
Why? Because it allows us to understand what are the
key determinants of productivity, , in the Solow model.
Recall that productivity is key to understand growth.

To this end, first find the expression for the general
Cobb Douglas production function in per capita terms:
Now we are going to introduce and adopt the
convention of small caps variables to represent per
capita variables. In other words, we write:
And the production function in per capita terms simply
becomes:
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This expression tells us something important: that
productivity in the Solow model depends on technology
A, the capital share , and physical capital per person .
Due to concavity, this per capita (or “per worker”)
production function has the property of diminishing
returns to capital (see the class slides for chapter 12 for
more details on the topic). The following graph illustrates
this property and how our per capita production function
looks like in practice.
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SOLVING THE MODEL: THE FUNDAMENTAL EQUATION
How can we study physical capital per person in more
details? We need to do some algebra. In particular, we
start by dividing the left hand side and the right hand

side of equation 8) by . This gives us the expression for
capital accumulation in per capita (or per worker)
terms.
Which becomes:
Now look at the left hand side of the expression above
for a second, we have sort of a “miss match in timing” on
the left hand side because the numerator of that
expression is future physical capital while in the
denominator there is current population so we need to
use a little trick. Nothing fancy but you need to be
careful. Rewrite that expression as follow:
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What was the trick? I just divided and multiplied the left
hand side of that expression by . That is the trick.
Simple.
At this point we also need to recall equation 5) that tells
us:
so we can rewrite it as
Using this fact we can finally write the expression for
capital accumulation in per capita (or per worker) terms
as:
So using the convention of small caps variables to
represent per capita variables we have seen above, the
expression now becomes:
Or:

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This expression is called the fundamental equation of
the Solow model. It is crucial for what we are going to
do next.
It describes the evolution of capital per worker over time.
In particular, this expression from the economic point of
view, suggests that:
Capital per capita (or per worker) increases with
increases in savings per worker.
Capital per capita is negatively affected by population
growth, since for k to properly grow we have to
compensate for the growth in population (labor force).
Capital per capita is negatively related to the
depreciation rate (d).
Ultimately whether physical capital per person will grow
or shrink will depend on the above mentioned forces.
In particular, if savings per worker are more than enough
to compensate for the capital needed for population
growth and depreciation (this is called the break even
capital), we have that capital stock per worker grows
through time
On the other hand, if savings per worker are not enough
to compensate for population growth and depreciation,
we have that capital stock per worker shrinks over time.
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What’s next? Finally, we can now make predictions for
the future of the economy. We can answer the following
question: will our economy stop growing? If so, at what

level of productivity/output/consumption/investment?
THE CONCEPT OF STEADY STATE
In general, we say that any (economic) variable is in
steady state when that variable does not grow over time:
the variable simply stays constant over time this
means that the value of a variable in period t is equal to
the value of the variable in period t+1 for any time t of
your choice. This is the concept that we are going to
adopt to study the long run equilibrium in this economy.
STEADY STATE IN THE SOLOW MODEL
Can capital accumulation by itself generate sustained
growth? Can income per capita grow at a positive and
relatively constant rate forever? The answer to this
question is NO for a simple reason: diminishing marginal
product of the factors of production (i.e., diminishing
marginal product of labor and capital). If an economy
doesn't improve its technology, it will not be able to
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achieve sustained growth just by increasing its capital or
its total labor.
How can we see this? In our fundamental equation of
the Solow model, when is the capital per capita reaching
the steady state?
Formally, the steady state for capital per capita happens
when:
Where is just a constant (we are going to figure out the
actual number behind that constant).

Now distribute the terms of the fundamental equation
of the Solow model and do some algebra:
With this latest expression and the mathematical
definition of steady state just given, we can study the
steady state of the Solow model, and graph the Solow
diagram.
If we plug
into the fundamental equation of the Solow model just
written above we get:
Solving this equation for :
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“small”, . I am aware we are actually making an
approximation mistake here with this assumption, but
most likely not a big one. Actually, this way we can have
a very rough idea about the magnitude of the error we
are making. If we were not to introduce this assumption
and commit this error we would have to draw a
tridimensional plot. This is easy with computers but not
so much on paper/board (at least not for me).
Now go back to the expression for capital accumulation
in per capita (or per worker) terms. It was:

If we now use our approximation on the right
hand side only we have an idea of the approximation
error:
The left hand side of this expression is a proxy for the
growth of physical capital in per capita terms (and for
the approximation error).
The first term on the right hand side is savings in per
capita terms (=investment in per capita terms).
The second term on the right hand side is the amount
of physical capital per person needed to replace the
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Note that the graph is just expressing in pictures what we
have done with the math previously. On the horizontal
axis there is physical capital in per capita terms.
There are 3 curves in the graph that you need to
familiarize yourself with:
I) The straight line from the origin in green is simply
This is called “break even investment” the reason why it
is called so is because it is the amount of new machines
per person that need to be bought/produced to make up
for all the machines that break down and for the extra
machines needed to keep up with the increase in
population (i.e. new workers coming into the economy).
II) The blue curve line starting from the origin that is
“higher” out represents total output per person

It is a concave function that has that shape in the
diagram because of the decreasing marginal return to
capital as we have seen previously.
III) The red curve line starting from the origin that is
“lower” in represents total savings per capita
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It has precisely the same shape as the curve representing
total output, but it is just sort of shifted closer to the
horizontal axis because it is total output is multiplied by s
(the saving rate) which is a percentage.
Graphical visualization of the steady state:
When the curve representing total savings
crosses precisely with the line representing “break even
investment” (i.e. the two expressions are
precisely the same) then we find the steady state of
capital per person of the economy
And, again just to reiterate this once more, with the
precise numerical value for you can then plug it in the
expression for output per capita and obtain the output
per capita of steady state
You can also easily obtain the savings per capita of
steady state .
And subtract savings per capita of steady state from
output per capita of steady state to obtain consumption
per capita of steady state

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Note 2 important things:
there are actually two points at which the curve
representing total savings crosses precisely
with the line representing “break even investment”
in the graph. The first point is, however,
the origin of the axis. Since it does not have a very
meaningful economic interpretation to have zero
capital per person and zero saving per person and
zero output per person, we disregard that steady
state. The second point of intersection is .
is called a stable steady state. This
means that if by any chance for whatever reason the
economy gets away from it, it does tend to go back
to it. Why? To see this, look at the graph and
suppose the economy is starting at a point to the
right of , on the horizontal axis. Just looking at the
graph, this implies that
This in turn implies also that, from the (modified)
fundamental equation of the Solow model:
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So if physical capital per person is
decreasing over time. Until when does this go on? Until
physical capital per person goes precisely back to
where .
The opposite happens when the economy is starting at a
point to the left of : the physical capital per person will
grow toward .

Bottom line: the economy reverts to always. And that’s
why we say that is a stable steady state.
IMPORTANT LESSONS FROM THE SOLOW MODEL
The sort of unpleasant conclusion of this model is that
there is a limit to growth and that limit is precisely
defined by (which in turn implies a limit to output per
person and consumption per person). When the
economy reaches the steady state, we are sort of stuck
there forever.
Note, however, that the model predicts that there is a
limit to growth of physical capital per person (and hence
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output per person and consumption per person) not to
actual physical capital and output. Why?
Well, think about it, if the economy gets stuck at a
certain constant, and since population is the
denominator of physical capital per person is growing at
the rate of n, then it means that physical capital which is
the numerator of that variable must be growing at the
same rate as the denominator, n to maintain that
constant level .
This also implies that total output is growing at the rate n
in steady state for the Solow model. And total
consumption as well. What’s next?
POLICY ANALYSIS

Can the government do anything to spur/affect growth?
Let’s see two examples.
Consider a policy that changes the saving rate s
will shift the total saving curve up or down
(depending on whether the policy implies an
increase or decrease in savings).
In the example below we increase the saving rate
from a level s1 to s2 new curve for “saving per
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depend on s (only on n), so there’s no change in the long
run (steady state) growth rate of the economy.
Per capita variables do not grow in steady state.
B) In the short run (= in the transition between the two
steady states), savings per worker are larger than
(n+d)*k, so the economy grows faster than usual. This
increased growth rate will be reduced as the economy
approaches the new steady state, until, in the end, we
are back to the same growth rate as before.
C) Nevertheless, per capita capital and income levels are
higher in the new steady state. This is the long run effect
of increased savings in the Solow model.
Consider a policy that changes the population
growth rate n will change the slope of “break
even investment” line.
In the example below we increase the population
growth rate from a level n1 to n2 new line for

“break even investment” in the graph below is the
one in dashed green we reach a new steady state
lower physical capital per person and output per
person than before ( .
Page 31 of 41
population growth rate, so there’s an increase in the
growth rate of aggregate variables.
C) In the short run, savings per worker are smaller than
“break even investment”, (n+d)k, so the economy grows
slower than usual. This means that, in the short run, the
growth rate of the per capita variables will be negative (y
and k falling). Again, this growth rate approaches zero as
the economy approaches the new steady state, until, in
the end, we’re back to the same growth rate as before.
D) Nevertheless, per capita income level is lower in the
new steady state. This is the long run effect of increased
population growth on the per capita variables in the
Solow model.
Lastly:
What about a policy that changes d? (think about this
yourself! Draw a graph and think about it!)
What about a policy that policy that changes A (think
about this yourself! Draw a graph and think about it!!)
Page 32 of 41

IMPLICATIONS AND PREDICTIONS OF THE MODEL
The Solow growth model, assuming everything else the
same, implies the following:,
Country with higher saving rate (s) enjoys higher
GDP per capita in the long run (look at the
expression of steady state or at the diagram).
Two countries with the same initial aggregate capital
stock (K), the country with higher saving rate grows
faster (look at the fundamental equation or at the
diagram).
Two countries with the same saving rate (s), the
country with lower initial aggregate capital stock
grows faster (look at the fundamental equation or at
the diagram).
Country with lower population growth rate (n)
enjoys higher GDP per capita in the short run and
lower GDP growth in the long run (look at the
fundamental equation or at the diagram).
Two countries with the same initial aggregate capital
stock, the country with lower population growth rate
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grows faster (look at the fundamental equation or at
the diagram).
In the long run, all the countries with the same

parameters (n, d, s, A, ), but different initial capital
stock reach to the same GDP per capita (look at the
expression of steady state or at the diagram).
In the long run, per capita GDP stops growing for all
countries (look at the expression of steady state or
at the diagram).
This last one is probably the most disappointing
prediction of the Solow model. In order to go beyond this
prediction that basically says that there is a limit to per
capita growth, economists have tweaked the Solow
model in a number of ways. You will probably study
some of them in intermediate Macro, if you plan to take
that course.
CHECKING THE MODEL AGAINST ESTABLISHED CROSS
COUNTRY FACTS
If you remember the scientific method it suggests that
one starts observing a phenomenon, then formulates a
theory (based on some assumptions), and then finally
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one needs to go back to the data to check whether the
theory works or if there is a need for a modification of
the assumptions made. Or if the theory is falsified.
Now, after we have formulated a theory for growth (the
Solow model) that has some predictions (the one we just
saw), we need to be looking at few cross country facts
about growth. Those are then compared with the
implications of the Solow growth model to see how the

model fares.
EMPIRICAL FACT 1: Data show that almost all the
countries are getting richer.
This is not an implication of the Solow growth model (at
the per capita output level). This is probably the most
criticized part of the basic model. The basic model could
be modified to fit this empirical fact though.
EMPIRICAL FACT 2: There is a
positive correlation between the investment and output
per worker across countries. This is consistent with the
Solow growth model. If the only difference across
countries is the saving rate s, a country with higher s
exhibits a higher level of output per capita.
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EMPIRICAL FACT 3: There is a
negative correlation between the population growth
rate and output per worker across countries.
This is also a basic implication of the Solow growth
model. If all the countries are the same except for
the population growth rate n, a country with higher n
exhibits lower level of output per capita in steady state.
EMPIRICAL FACT 4: Countries with higher saving rates
have higher capital output ratios. This is consistent with
the Solow growth model. If the only difference across
countries is the saving rate s, a country with higher s
exhibits a higher level of physical capital as a fraction of
output.

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THE ABSOLUTE AND RELATIVE (or CONDITIONAL)
CONVERGENCE DEBATE
EMPIRICAL FACT 5: There is no correlation across
countries between the level of output per worker in
1960 and the average growth rate of output per worker
during the period 1960 2000. However, there is a
negative correlation among the richest countries.
Page 37 of 41
Page 38 of 41
One of the most important implications of the Solow
growth model is that, if the only difference among
countries is the initial stock level of capital, the level of
output per capita across countries will keep shrinking
and eventually all the countries should enjoy the same
level of output per capita. Please see the figure below.
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Output per capita (in log)
US

Country A
Country B
Time
This feature depicted in the graph above is called
absolute convergence. Unfortunately, the data suggest
that the convergence is not happening among all the
countries, but we see convergence among richest
countries (G 21 countries). So the question is, why the
convergence occurs for some countries and not for
others.
One potential explanation to this puzzle is that the
countries which are converging are the ones which have
similar underlying economic conditions (in terms of s, n,
d, A, and ), but those which do not experience
convergence to the US are the ones which are different
from the US in terms of those underlying economic
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Assume country 1 and 2 have similar parameters (A, s, n,
d, ) to the US (but different initial levels of output per
capita and capital per capita), but country 3 differ in the
level of s. In other words, in country 3 something other
than the initial stock of capital is different from the
fundamental parameters of the US (and country 1 and
2). Note that, even though country 3 starts very close to
country 2 in terms of output per capita (and capital stock
per capita), it converges to a different steady state than
country 2 because of this difference in the saving rate.

Country 3 converges to a lower level of output per
capita, but not to the US level of output per capita, while
country 1 and 2 will eventually converge to the US level
of output per capita.
This idea is called conditional (or relative) convergence:
even if countries may differ across the initial level of
output (or physical capital) per capita, they converge to
the same steady state because they have similar
fundamental parameters (A, s, n, d, ) in their economy.
Countries that may have similar initial level of output (or
physical capital) per capita converge to a different
steady state because they have different fundamental
parameters (A, s, n, d, ) in their economy.
notesNotes on Solow Model tutor

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