1
1. Review of Solow Model
1.1
2.1.
Stylized Facts
Basic Solow Model Assumptions/Simplifications
Two Key Equations:
Production Function
Capital Accumulation Equation
Solving the model with diagrams
Closed form solution of the steady-state
Properties of the steady-state
Comparative Statics
An increase in the saving-investment rate
An increase in population growth rate
Transition Dynamics
2.2
Solow Model with Exogenous Technology
Notice that in the model without technology, per capita output and per capita capital do not
grow at the steady-state. This is not consistent with empirical data because we know that most
advanced economies such as the US exhibit growth in per capita variables in the long run.
Thus the prediction of the model contradicts the empirical data in positive per capita growth.
In order to produce growth in per capita variables, Solow has added a technology variable A
into the model. This variable enters the production function as follows:
Y = f (K, AL) = Kα (AL)1-α
In this production ft., labor is more productive when the level of technology is higher. We can
call AL “effective labor”.
One important point is that Solow assumes that technological progress is EXOGENOUS.
Technology being exogenous means that no matter what the agents do in this economy,
technology variable A always grows with the rate g: BY ASSUMPTION,

A
 g  A  A0 e gt
A
This means that he does not explain in the model HOW technology is improved. We will try
to understand how technology is improved in later chapters but now this is the simplest way
to include technology.
The capital accumulation eqn. is the same as before:
2


K
Y
K  sY  K   s  
K
K
(I)

K
where
is the growth rate of capital stock.
K
Growth Implications with Technology:
Consider (I): K grows at a constant rate if and only if
Y
Y
is constant. For
to be constant,
K
K
both Y and K must be growing at the same rate.
Y
Y
y
Y
y
Notice also that
is constant,
is also constant: y and k will be growing
 L  so if
K
k
K K k
L
at the same rate. Denote the growth rate of y as gy and the growth rate of k as gk . In the above
scenario, gy = gk .
A growth scenario where capital, output, consumption and population grow AT A
CONSTANT RATE at the steady-state is called a balanced growth path.
This is a desirable output for a model because this is consistent with the empirical data of the
US and many other developed nations (advanced economies).
Now, write the production function in terms of y 
Y  K  A1 L1 
Y
 K  A1 L  y  k  A1
L

Take time derivative

Y
K
and k  .
L
L
Take log s log y   log k  (1   ) log A

y
k
A
    (1   )
y
k
A
Then, g y  g k  (1   ) g . Since gy = gk on a balanced growth path,
(1   ) g y  (1   ) g  g y  g  g k
We have shown that output per capita and capital per capita grow at the rate of technological
progress g. In the previous model without technology, we did not have growth in per capita
variables y, k, consumption = (1-s)y . After introducing the technology variable, we are able
to obtain positive growth in these variables. This means that technological progress is the
source of sustained (long-run) growth in per capita output, capital and consumption.
The Solow Diagram with Technology
3
~ K
In this version with technology, we define a new state variable, k 
, capital per effective
AL
labor. A state variable in a model is supposed to be constant at the steady-state by definition.
~
That is why we have chosen k as our state variable.
Notice that on a balanced growth path, k 
~ k
K
grows with g, k  is constant, since A
L
A
Y
y
y
 is constant.
grows with g too. Also, y grows with g, and ~
AL A
~
Now, rewrite the production ft. in terms of k and ~y :


Y  K ( AL)
1
~
Y
 K 


y k
 ~
AL  AL 
~
Now, rewrite the capital accumulation eqn. in terms of k and ~y :



~
~ K
k K A L
First, note that by k 
, take logs and differentiate wrt time we get, ~    .
AL
k K A L
Plug this into capital accumulation eqn:
~
k
Y
~  s   g  n 
K
k
~
~
~
~
k
y

s

(
n

g


)

k
 s~
y  (n  g   )k
~
~
k
k
Now, we can draw the last expression by separating hte first term s~y and the second term
~
(n  g   )k . SHOW FİGURE 2.9 FROM THE COMPUTER. Notice that this graph looks
~ k
very similar to the one with the Simple Solow model. But remember that k 
and
A
Y
y
~
~
y
 , these are different with the simple model. If the economy starts from a k that
AL A
~
~
is smaller (larger) than the steady-state level, then k grows (declines). This continues until k
becomes equal to its steady-state level.
~
~
When k is constant at the steady-state, this means k is growing with rate g. When k is
growing, this means that k is growing faster than g. The same thing happens for output:
When ~y is constant at the steady-state, this means y is growing with rate g. When ~y is
growing, this means that y is growing faster than g.
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~
Can we solve for k  in a closed form solution? We can from the capital accumulation eqn.
~
~
~
~
.
Plug
in
k
 0 and we get
k  sy  (n  g   )k

~ 
s

k   
n

g




1
1
and

s
~y   
 n  g   



s
y
y  , y  (t )  A(t )
Since ~
A
 n  g 




1

1
According to the above solution, the implications of Solow model with technology is similar
to the implications without technology. A larger saving rate implies a higher level of output
per capita: Also, a larger rate of population growth implies a smaller level of output per
capita. But more interesting result is:
Can the Solow model produce a change in the long-run growth rate? That is, can the
Turkish government change Turkey’s long-run growth rate by some kind of policy like
subsidizing investments, by giving tax breaks for investments? According to the Solow
model, NO. Because the Solow model says that any effect of an increase in the saving rate
TEMPORARILY produces an increase in the growth rate of output per capita y.
Let us illustrate the effect of a policy of an investment subsidy by the Turkish government.
This will increase the saving rate from s to s’. As Figure 2.10 SHOWS, this will shift the
gross savings curve upwards. Then, at the current level of capital per effective labor, there is
~
more investment than the level of investment needed to keep k constant. During the
~
~
transition period, this causes k to increase until k ** , the new steady-state level (this does
NOT happen instantly). The transition dynamics can be observed on Figure 2.11.
~
~
k
y
 1
~  s ~  (n  g   )  sk  (n  g   ) is positive during the transition to the new steadyk
k
state. What does this mean for y ? Does y keep growing at g during transition period? No, y
grows faster than g during transition. Notice that
Y
y
~
y
 ,
AL A
~
y  A~
y  Ak  ,
~
y
k
 g   ~ . This means that if
y
k
~
k
~ is positive, then capital
k
per labor grows faster than g. This is what we observe on SHOW Figure 2.12 and 2.13. At the
steady-state, when the transition period ends, the growth rate of y and k go back to g, growth
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rate of technology. And the government cannot change g with any policy. The effect of
government policy on growth rates is only temporary in the Solow model.
How can we explain differences in growth rates across countries in the context of Solow
model? One way is to appeal to differences in technological growth rates g in different
countries. So countries with high technological growth has a higher economical growth. But
this interpretation has a problem because we are assuming that technology consistently grew
faster in South Korea than the US for the last 40 years. Why? This cannot be very plausible.
Another possible explanation is to appeal to transition dynamics. We know that the saving
rate has increased steadily in Singapore, South Korea and Taiwan. Show Figure 2.14.
Remember that output per labor grows faster than the technological growth during the
transition period when the economy is trying to get to the new higher steady-state. So we can
explain the fast growth of the Asian economies by saying that their steady-state output per
labor keeps increasing fast because of increasing saving rates. Everytime their saving rate
increases, the distance between the current y and the steady-state y gets larger. This causes the
y to grow at a rate much larger than the technological growth. This also explains the rapid
growth of Germany and Japan after WWII. Because their initial y & k were close to zero, but
their steady-state income level was high.
Growth Accounting
In 1957, Solow wrote another paper in which he broke down growth in output into growth in
capital, growth in labor and growth in technological change. In this case, he proposed
Y  BK  L1
this production function decomposes growth into three pieces:
Y
K
L B
   1    
Y
K
L B
Here B is called Total Factor Productivity. If we are more interested in output per labor y
instead of total output Y, then we modify the above equation by subtracting
sides. Notice that
L
from both
L
K L k
Y L y
y
k B
  . Then,    . SHOW TABLE 2.1
  and
K L k
Y L y
y
k B
Generalizations on the table:
1- output per hour worked instead of output per labor.
2- an additional term about changing labor composition. changing education level,
3- divide capital into information technology and other services.
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There is a slowdown of growth between 1973 and 1995. Show the same pattern in Turkey
with the Excel file. This is explained by the productivity slowdown in these years. Why did
the productivity fall? Possible explanations:
1- Declining effect of WWII inventions on the private sector,
2- Oil crisis of 1973-74,
3- Shift from manufacturing sector into service sector.
Faster growth after 1995, both in Turkey and in the US. Why? Some people in the US say the
use of PCs in production has improved productivity a lot.
Explain growth of Asian tigers using growth accounting approach: SHOW FIGURE 2.15.
Notice that all Asian economies are above the 45 degree line. This means their TFP growth
rates were smaller than the growth rate of output per labor. Their growth was more due to:
1- physical capital accumulation,
2- better education
3- higher labor force participation
4- shift from agriculture toward manufacturing sectors.
Compare with the Solow model: B  A1 There is a similarity between the concept of TFP
and technology.
CHAPTER 3
Empirical Applications of Neoclassical Growth Models
Solow Model with Human Capital: Mankiw, Romer Weil Approach
Gregory Mankiw, David Romer and David Weil improved the fit of Solow model by
adding human capital into it.