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The Difference Betweeen Level and Growth Rate of a Variable
Some of you have a problem with distinguishing the two. THERE IS A DIFFERENCE
between LEVEL and GROWTH RATE of a variable. For example, level of GDP Per
capita (GSMH) in Turkey was $2,598 in 2002, $3,383 in 2003 and $4,172 in 2004.
Between 2002 and 2003, growth rate of GDPPC was 30% = (3383-2598)/2598. Between
2003 and 2004, growth rate was 23% =(4172-3383)/3383. Growth rate of a variable is
the percentage change of the level of that variable during a certain period of time.
In the models, y is the level of GDP per capita. k is the level of capital per capita. Growth
rate of GDP Per Capita is calculated by
y
. y means we are taking the change in the
y
variable during one instant of time. If we divide this value by the level of output per
capita, we get the growth rate in one instant of time. In the real examples of annual
growth rate, growth rate is calculated during one year, a long time compared to an instant,
but the logic is the same. Take difference for a given period of time, and then divide by
the level. For example, Solow model finds that the steady-state level of y as:

s
y  (t )  A(t )
 n  g 
The growth rate of y is

 1


y
y
. In the Solow model, (growth rate of y) =
is equal to
y
y
growth rate of technology, g, a constant.
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Chapter 8: The “AK” Model
An important discussion among economists is: “Can the government policy
increase the long-run growth rate of the economy?” Some people say it can, some people
say it cannot. Government policy can subsidize research, education and investment. In
the Solow and Romer models, we found that government policy cannot increase the longrun growth rate of the economy. Government policy can change the growth rate only
temporarily. After some time, the growth rate goes back to its initial level.
Some people think government policy must be able to increase the long-run
growth rate of the economy. Today, we will cover a model that allows the government to
increase the long-run growth rate of the economy: the AK Model. In this model, when
government subsidizes investment, education or research, the long-run growth rate
increases permanently.
ELEMENTS OF THE MODEL:
Remember the original Solow model. Assume there is no technological process.
Also, change the production function so that   1 :
Y  AK
where A is a positive constant.
This is why this model is called “AK” model. The capital accumlation equation is the
same as in Solow:

K  sY  K
For simplicity, assume that population size is one, L=1, and population does not grow,
n=0. This means that we can use K for capital per capita (k before) and Y for output per
capita (y before).
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

Since Y  AK , K  sY  K becomes K  sAK  K . Here, both s and A are constants.
So if we draw the diagram, it will be like the following: Draw Figure 8.1. Notice the
difference with the original Solow diagram. The function that gives gross investment
sY=sAK is a linear function of K. Then it never crosses the K curve. This means that
starting from any initial level of capital per capita, capital per capita keeps growing for an
infinite amount of time. It never stops growing. In other words, there is NO STEADYSTATE level of capital per capita in this model.
Remember in the original Solow model, gross investment was skα , a concave function of
capital per capita. The difference with the AK Model is that in the AK model,   1 . This
makes the gross investment function LINEAR. Then, gross investment function never
crosses depreciation function K .
Growth Implications:
The growth implications of the AK model is fundamentally different from the Solow
model. I will now show you that increasing the investment rate increases the growth rate
of the economy permanently. This is IN CONTRAST with the Solow model. In the
Solow model, increasing s increases growth rate only temporarily, during the transition
period. After the transition period ends, growth rate becomes equal to the technological
growth rate g.
In the AK Model, Y  AK . Take logs: log Y (t )  log A  log K (t ) . Take derivative with
respect to time:
Y K
 . The first term disappears because A does not change over time,
Y K
it is a constant.

Now, from the capital accumulation eqn., K  sAK  K divide both sides by K:
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
Y K
Y
K
We get
 sA   . Then, by  , we get  sA   . This means that the growth
Y K
Y
K
rate of the economy is always equal to sA   . This is different from Solow model where
growth rate was equal to g and exogenous. Government policy cannot change g. But here
growth rate depends on s. If s increases, growth rate increases. This means that if
government can subsidize investment by giving tax break for investors, then government
can PERMANENTLY increase the economy’s growth rate. This is the key property of
the AK model. Government policy can permanently change growth rate of the economy
in the AK Model.
Technological Development: Intentional Research or Learning By Doing?
There are two ways to model technological progress in a growth model. One way is to
assume there are a group of researchers working ONLY for developing technology. This
is the way the Romer model approaches technology. Romer model assumes that part of
the population works only as researchers and develop new technologies.
There is a second way to model technological development. This approach sees
technological development as a by-product of economic development. This is also called
“learning by doing” in the literature. According to this approach, as workers and
professionals increase their experience, they can develop new technologies as a byproduct. For example, an experienced lathe operator (torna ustasi) can build a new lathe
with better performance. Or a car mechanic can invent a better technology for the
automobile engine. Their job is not to invent new technologies but they just learn by
doing. Do you think this is how technology develops or do you think there must be
people professionally working only for research?
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Mathematically, it is easy to model the “learning by doing” approach. Consider the
production function the same as in the Solow Model:
Y  BK  L1
Now, assume for simplicity that L=1. Then, Y  BK  .
Here, B can be seen as the level of technology. Each firm takes the level of technology B
as a given constant. This is because each firm is small compared to the total economy.
There are many firms in the economy. However, when we take total-aggregate economic
activity into account, economic activity increases the level of technology as a by-product.
Technology develops as a result of learning by doing. For example, a larger capital stock
can increase the level of technology. Mathematically, we can write such a technology
function as:
B  AK 1
where A is a constant number. So if capital increases, technology level
increases.
Remember our discussion about positive externality of research. Individual firm does not
accumulate capital because it develops technology, firm accumulates capital because it
must use it in production process. Firm ignores the positive technology effect. But this
capital accumulation has a positive by-product, an externality: it increases the level of
technology in the economy. So there is a difference between private returns of capital
accumulation and social returns of capital accumulation. Society benefits from capital
accumulation, but individual firms do not take this effect into account.
This is the idea of learning by doing approach. When we plug this technology function
into the production function, we get:
Y  BK  , Y  AK 1 K  , Y  AK .
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We get the AK model as a result. So, this is an alternative way to model technological
progress.