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Chapter 5: Endogenous Technological Change
In 1990, Paul Romer has written a paper with the same title. His contribution is to
model how technological progress occurs. Remember that in the Solow model
with technology, technology grows exogenously. This means that the technology
grows only because we assume that it grows. There is no factor inside the model
that causes technological progress. There is no mechanism. Paul Romer
naturally objected this kind of a simplistic approach. A good model should have a
mechanism of technological progress in it. Romer’s approach is that part of the
labor force works in the Research and Development sector. These researchers
search for new ideas. Those new ideas increase the stock of knowledge and
ideas A in the economy. As a result, other factors capital and labor becomes
more productive. This is how technology progresses in Romer’s economy.
5.1
Elements of the Romer Model
Formally, Romer proposes the following production function:
Y  K   ALY 
1
(1)
Notice that this production ft. is very similar to the one we had in the Solow model
with technology EXCEPT the LY , the part of labor that works in production.
Romer assumes that LA part of labor force works as researchers and the
remaining labor works in production. For simplicity, we assume the fraction of
population that works in the research sector sR 
LA
is constant. (In the original
L
Romer paper, s R is endogenous). Of course, LY  LA  L and
LY  1  sR L .
Similar to Solow Model, we assume that population grows at a constant rate n:
2
L
n
L
Also, notice that Romer considers A as the stock of ideas or knowledge. Stock of
knowledge is assumed to be an endogenous factor. This means he proposes a
mechanism of how stock of knowledge grows over time. This means he proposes
a production function for ideas. In particular, he assumes
A   LA
(2)
Where LA is the number of workers working in the research sector. The stock of
knowledge increases if there is more people working in the research sector.
 is the productivity of one researcher. One more researcher increases the stock
of knowledge by  units.
An individual researcher takes productivity of a researcher  as given. This is
because one researcher is very small compared to the population so she cannot
change the value of  by herself. But for the aggregate economy, productivity
may depend on the existing stock of ideas. For example, the invention of the
number zero, the algebra, the radio, the computers made today’s researchers
more productive. For example, without Isaac Newton’s findings about gravity,
many findings about space would be impossible. Newton’s ideas “spilled over” to
help future researchers. Let us call this effect “spillover effect”. This means that
productivity of research  should be an increasing ft. of the current stock of
ideas A.
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On the other hand, some people argue that the most obvious ideas are
discovered first. It becomes harder to find new ideas. This is like saying catching
fish should be harder if other people have already caught a lot of fish on Galata
Bridge. Let us call this argument the “fishing out” argument. This argument says
that research productivity should be a decreasing ft. of the current stock of ideas.
WHICH ONE DO YOU THINK MAKES MORE SENSE?
We need a function that allows both of these possibilities. We can use:
   A
where  is a constant.
(3)
If   0 , then research productivity is an increasing ft. of the current stock of
ideas. This is the spillover argument. If   0 , then research productivity is a
decreasing ft. of A. This is the fishing out argument we have discussed. If   0 ,
then productivity is independent of the current stock of ideas.
Now, the stock of ideas evolves according to
A   A LA
(4)
Remember that an individual researcher is taking  as a constant. This is
because her research effort is very small compared to the total stock of
knowledge A. Her actions cannot change A significantly to change  . However,
when we think of the aggregate research effort, total stock of knowledge might
have a positive “spill over” or a negative “fishing out” effect on the production of
new ideas. This is obvious by (4) above.
The capital accumulation equation is the same as in the Solow model:
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
!!!!!CORRECTION: K  sK Y  K
(5)
Growth in the Romer Model:
Notice that following the same arguments in Solow Model with technology,
growth rate of y and k is equal to growth rate of A on a balanced growth path.

Using the capital accumulation equation, K  sK Y  K , we know that for K to
have a constant growth rate on a balanced growth path, Y/K must be constant.
Then, y/k must be constant. Then, y and k must grow at the same rate: gy = gk.



y
k
A
Next, we use the production function to show that:    (1   ) and that
y
k
A
g y  g k  (1   ) g A . It follows that g y  g  g k .
The above result shows that if there is no technological growth, there is no
growth in the economy. This is similar to the Solow result. But the difference
btw Romer and Solow is that we are able to determine the growth rate of
technology in terms of other parameters. This means that we are able to
link the growth rate of the economy to factors such as population growth.
Consider (4): A   A LA . Divide both sides by A:
A  LA

.
A A1
On a balanced growth path, growth rate of technology gA must be constant.
Then, the right hand side must also be constant. If we take logs of both sides and
take derivative,
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g A L A
A
A L

 (1   ) . The left hand side is zero, therefore: (1   )  A .
g A LA
A
A LA
Notice also that on a balanced growth path, growth rate of number of researchers
must be equal to the growth rate of the population (since we assume s R 
LA
is
L
constant).
Then, g A 
n
. The rate of technological growth and hence per capita income
(1   )
growth is closely related to the population growth rate. Notice how the
interpretation of population growth has changed btw Romer’s model and the
Solow model. In the Solow model, faster population growth meant a lower level
of income per capita (makes you poorer). Similar negative effect of population
growth on the steady-state level of income per capita also holds in the Romer
model. However, In the Romer model, population growth is the only source of
technological progress and any long-run growth in per capita income (now it is a
good thing!). A higher population growth in the Romer model means a higher
long-run economic growth rate.
Can we test this result empirically? SHOW Figure 4.4. World Population growth
rate is close to zero (0.075 % on average) before the 18th century. It started to
increase in the 18th century. In the last 40 years, world population growth was
around 2% annually. This is a huge increase compared to before 18th century.
Notice that the Industrial Revolution and the rapid economic development in
Europe and other countries of the world started around the same time, the 18th
century. So this empirical observation supports the prediction of the Romer
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model: Long-run Per Capita Income growth is a consequence of population
growth.
The following report for year 2003 is taken from US Census Bureau website
www.census.gov/ipc/prod/wp02/wp-02003.pdf , the American institution that is in
charge of following population movements in US and the World: (on page 13)
“...At the opposite end of the spectrum, more than half of the world’s more
developed countries (MDCs), including those in Eastern Europe and the former
Soviet Union, are expected to experience population declines over the next 50
years....”
If Romer’s prediction is true, what is expected to happen to Eastern European
and the former Soviet Union economies in the long-run? Are they expected to
grow or decline over the next 50 years?
Another implication of the Romer model is similar to Solow Model: Government
cannot change the growth rate of per capita income by ANY policy EXCEPT a
policy to increase the fertility rate. For example, a policy of increasing the
saving/investment rate sK , or increasing in the R&D share of labor sR cannot
change the long-run growth rate of the economy. This is because these
parameters do not appear in g A 
n
. They only increase the growth rate
(1   )
during the transition period. They also increase the LEVEL of income. We will
look at this in more detail below.
Therefore, making economic growth endogenous does not allow policy makers to
manipulate long-run growth rate. One exception is the incentives given to families
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to have more children. If that policy can increase the population growth rate, then
economic growth rate can increase according to Romer’s model.
Exercise : To increase sR 
LA
L
Analyze the short-run and long-run effects of a Policy to increase the R&D share
of labor sR . Suppose sR increases permanently to s’R because say the
government increases support funds for research activities. Analyze this question
in context of Romer model with   0 .
Notice that in Romer model, the part of the model that governs technological
progress is separate and independent from the part of the model that governs
capital and income growth. Once we determine what happens to technology in
the Romer model, the analysis of what happens to income per capita is the same
as
we
did
in
the
Solow
model
with
technology.
Technological
Progress
Romer
Model
Capital &
Income Growth
Solow model
with Technology
First, let us consider what happens to technological progress. Check the
production function for ideas with   0 : (Recall original A   A LA ) =>
A   LA
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Divide both sides by A and we get
write
L
L
A
  A . By s R  A  L A  s R L , we can
L
A
A
A
L
  s R . Notice that in this case, growth rate of technology equals
A
A
population growth rate at the steady-state: g A  n . Otherwise, growth rate of
technology cannot be constant. FIGURE 5.1 shows the effect of the change in
the R&D share. DRAW the figure again writing the horizontal axis as s R
the
function
g A   sR
L

A
as
sR
A
L
  sR
A
A
.
Notice
that
at
the
L
and
A
steady-state,
L gA

. That is, at the steady-state, the ratio of the number
A 
of technology workers to the level of technology
g
LA
equals A as seen on
A

graph.
At time t = 0 when the policy takes effect, L = L0 and A = A0 . But sR has jumped
to s’R . Then, we jump to the new point with a higher level of s R
L
L
 s' R 0 .
A
A0
The function that determines growth rate of technology (
A
L
  s R ) is an
A
A
increasing function of s R
L
. Therefore, in the short-run, growth rate of
A
technology will increase to point X. This means technology increases faster than
the population.
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But notice that when A increases faster than L,
L
decreases over time and by
A
A
L
  s ' R , growth rate of technology becomes smaller (Note that when R&D
A
A
share has increased once, it does not change anymore). Growth rate of A gets
smaller until it becomes equal to population growth rate. After that,
and equal to the population growth rate. The motion of
A
is constant
A
A
and log A are
A
illustrated on FIGURES 5.2 and 5.3.
Now what happens to income per capita level in the long-run can be solved using
Solow model equations, because the long-run income dynamics is (i)
independent from technology dynamics, (ii) equivalent to Solow model, in the
Romer model. We have found in Solow model that:

s
y  (t )  A(t )
 n  g 

 1

at the steady-state.

This result slightly changes in the Romer model as:

s
y  (t )  A(t )
 n  gA 

 1
 1  s R 

(6)
The only difference is the term 1  sR  because part of the population is working in
the research sector, so they cannot work in production.
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We know that the level of technology A increases when R&D share of labor
increases. Because, by g A   s R
L
s L
, we can write A(t )  R . Plug this into (6)
gA
A
and we get

s
y  (t )  
 n  gA 

1
s
 1  sR  R Lt 
gA

(7)
There are two opposite effects on per capita income when s R increases. The
positive effect on technology level increases the long-run level of per capita
income. But there is also a negative effect coming from 1  sR  . This is because
when there is more people working in the research sector, there is less people
working in production sector. Then there is less output and income as a result.
Which of the two effects dominate is not clear.
Increasing Returns to Scale in Technological Progress
Notice the effect of POPULATION SIZE L(t) on per capita income in equation (7).
Romer model suggests that a country with a larger population should have a
larger per capita income. This means, if Turkey has 70 million population and
$6,000 GDP per capita today, when it becomes 140 million people, it will have
$12,000 GDP per capita (this means TOTAL GDP is four times the old value,
from $420 billion to $1,680 billion). This is because of the way Romer formulates
nature of inventions. His formulation of technology is based on increasing returns
to scale in R&D activities. What does IRS mean? For example:
If 100 people discover 1 invention in one year, then 200 people discover strictly
more than 2 inventions in one year. This is the meaning of IRS. This is actually
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why government and charitable organizations (vakiflar) fund education and
research activities, because private markets do not produce enough research.
The social returns to research is always greater than private returns. When there
is such a difference between social returns and private returns, we say that there
is a POSITIVE EXTERNALITY from research.
Price of
research
Private Cost
social value
D-private value
Qmarket
Qopt
Qty of research
In this case, without government, private incentives lead to an outcome where
too little research is produced. To illustrate this, consider Newton’s inventions.
Did Newton get enough FINANCIAL reward for his HUGE contribution to
science? No.
PATENT SYSTEM is a mechanism to protect inventors’ interests by preventing
or excluding others from making, using, selling, offering to sell or importing the
claimed invention.
But it is still not enough for people to get sufficient benefit from making research.
Because of this, government and charitable organizations support researchers
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financially. Government and charitable organizations corrects the externality
problem by bringing private value of research close to its social value.
DISCUSSION QUESTION:
Do you believe it has been ONLY financial rewards that drive people to work in
scientific research in history? Or are there non-economical reasons for
inventions? Marie Curie discovered radioactive elements Radium and Polonium
in 1902. In an unusual move, Curie intentionally DID NOT PATENT the radium
isolation process, instead leaving it open so the scientific community could easily
research. She died because of exposure to the radioactive material. Can you
explain her motivation by pure economics? How can you explain it? To be
famous, or for service to the home country maybe. Non-material considerations
seem to become relevant here.
Exercise : To increase sR 
LA
L
Analyze the short-run and long-run effects of a Policy to increase the R&D share
of labor sR . Suppose sR increases permanently to s’R because say the
government increases support funds for research activities. Analyze this question
in context of Romer model with   0 .
Notice that in Romer model, the part of the model that governs technological
progress is separate and independent from the part of the model that governs
capital and income growth. Once we determine what happens to technology in
the Romer model, the analysis of what happens to income per capita is the same
as
we
did
in
the
Solow
model
with
13
technology.
Technological
Progress
Romer
Model
Capital &
Income Growth
Solow model
with Technology
First, let us consider what happens to technological progress. Check the
production function for ideas with   0 : (Recall original A   A LA ) =>
A   LA
Divide both sides by A and we get
write
L
L
A
  A . By s R  A  L A  s R L , we can
L
A
A
A
L
  s R . Notice that in this case, growth rate of technology equals
A
A
population growth rate at the steady-state: g A  n . Otherwise, growth rate of
technology cannot be constant. FIGURE 5.1 shows the effect of the change in
the R&D share.
DRAW the figure again writing the horizontal axis as s R
L
and the function as
A
A
L
L
. Notice that at the steady-state, g A   sR 
  sR
A
A
A
sR
L gA

. That is,
A 
at the steady-state, the ratio of the number of technology workers to the level of
technology
g
LA
equals A as seen on graph.
A

At time t = 0 when the policy takes effect, L = L0 and A = A0 . But sR has jumped
to s’R . Then, we jump to the new point with a higher level of s R
L
L
 s' R 0 .
A
A0
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The function that determines growth rate of technology (
increasing function of s R
A
L
  s R ) is an
A
A
L
. Therefore, in the short-run, growth rate of
A
technology will increase to point X. This means technology increases faster than
the population.
But notice that when A increases faster than L,
L
decreases over time and by
A
A
L
  s ' R , growth rate of technology becomes smaller (Note that when R&D
A
A
share has increased once, it does not change anymore). Growth rate of A gets
smaller until it becomes equal to population growth rate. After that,
and equal to the population growth rate. The motion of
illustrated on FIGURES 5.2 and 5.3.
A
is constant
A
A
and log A are
A