Endogenous Growth Theory
1.
The AK Model
AK1
AK2
AK3
Y  AK ,
K '  sY  K ,
L' / L  n
A0
0  s  1,
n0
0   1
Endogenous(3): Y , K ' , L'
Predetermined(2): K , L
Exogenous(4): A , s ,  , n
Production is proportionate to capital. In particular, there are no diminishing returns to
capital.
2.
Intensive Form AK Model
Define k  K / L and y  Y / L . Dividing AK1 by L we obtain
IF1
y  Ak .
Dividing AK2 by K , we obtain K ' / K  sy / k   . We want to eliminate K ' / K from this
equation. Taking the natural log of k  K / L and differentiating with respect to time yields
k ' / k  K ' / K  L' / L . Assuming L' / L  n , and rearranging terms, we have K ' / K  k ' / k  n .
Using this result, we can rewrite the equation K ' / K  sy  k as k ' / k  n  sy / k   . Multiplying
through by k yields
IF2
k '  sy  nk  k
Using IF1 and IF2, we may classify the variables in the intensive form model as follows:
Endogenous(2): y , k '
Predetermined(1): k
Exogenous(4): A , s ,  , n
3.
Analyzing the Intensive Form
Substituting IF1 into IF2, we obtain k '  sAk  nk  k , so
AIF1
k '  sA  n   k
Equation AIF1 is a linear homogenous differential equation with state variable k .  k  k ' / k
denote the growth rate of the per capita capital stock k . Equation AIF1 indicates
AIF2  k  sA  n    .
Real world economic data for developed countries indicates that per capita capital stock
grows at roughly a constant rate. The data also indicates that per capita output tends to grow at
roughly a constant rate, and at about the same rate as per capita capital stock. Differentiating
y  Ak with respect to time yields y ' / y  A' / A  k ' / k . So, to fit the data, the parameter A must
remained constant. That is, A' / A  0 must hold so y ' / y  k ' / k .
Assume A' / A  0 , so y ' / y  k ' / k . Define  y  y ' / y to be the growth rate of per capita
output. Using equation AIF2 and the result y ' / y  k ' / k , our model indicates
AIF3  y  sA  n    .
Real world economic data for developed countries also indicates per capita output and
capital stock levels grow faster than the employment level. The employment level growth rate is
given by  L  L' / L  n . If follows that the difference between the growth rate of per capita output
and the growth rate of employment is  y   L  sA  n     n  sA    2n . In order for this
difference to be positive, which is what we observe in the real world, we obtain the following
restriction on the model’s parameters
AIF4
A
2n  
.
s
Since A  Y / K is the average product of capital, the restriction AIF 4 indicates that the marginal
product of capital must be “high” in some sense, or high enough, in order for output and capital per
capita to grow faster than employment.
The growth rates  y and  k depend upon the values of the exogenous variables in the
model. Increases in either the savings rate s or the average product of capital A lead to higher
growth rates  y and  k . Conversely, a higher population growth rate n or higher depreciation rate
 leads to a slower rate of economic growth. In summary, the AK model indicates saving and
productive capital are the keys to obtaining a higher rate of economic growth, while population
growth and a higher depreciation rate hampers growth.
This AK model has advantages. One advantage is its simplicity. The primary advantage is we have
a model of endogenous technical change. There is no technical change assumed here. In particular,
the parameter A is held fixed. The growth in the productivity of labor, as measured by the growth
in the per capita output level y , comes when saving is high enough and when capital is productive
enough.
A disadvantage is that the model does not have a convergence property, which real world
economies seem to follow. That is, the growth rate of per capita capital stock does not slow in this
AK model as more capital is accumulated. For this reason, economists have sought other ways to
model endogenous technical change.
4.
GAK1
GAK2
GAK3
A Generalized AK Model
Y  AK  K  BL 
K '  sY  K ,
L' / L  n
1
Endogenous(3): Y , K ' , L'
Predetermined(3): K , L
Exogenous(4): A , B , s ,  , n , 
,
A  0,
0  s  1,
n0
B0
0   1
5.
Intensive form of Generalized AK Model
Let k  K / BL and y  Y / BL , then equations GAK1-GAK3 imply
y  Ak  k  , A  0 ,
k '  sy  k  nk ,
L' / L  n
GAK1
GAK2
GAK3
B0
0  s  1,
n0
0   1
Endogenous(3): Y , K ' , L'
Predetermined(3): K , L
Exogenous(3): A , B , s ,  , n , 
6.
Analyzing the Generalized Intensive Form Model


Together, equations GAK1 and GAK2 imply k '  s Ak  k   k  nk . Letting  k  k ' / k ,
we can divide this last equation through by k to obtain
AGAK1  k 
s
k
1
 sA  n    .
Assume sA  n   , and define
 k  sA  n    . It follows that  k   k  s / k 1 , which is positive. This indicates that, when the
per capita capital stock level is smaller, its growth rate is larger. Over time, the per capita capital
stock grows and the growth rate decreases, asymptotically approaching the level  k . Thus, this
model predicts that a less developed economy will have a higher growth rate, which is one way of
describing the conditional convergence property.
Equation AGAK1 exhibits the conditional convergence property.
The CES production function can generate both conditional convergence and economic
growth. This result is not worked out here, but obtaining this result is a good challenge.
7.
Research and Development Model
RD1
Y  1  a K K  A1  a L L 
RD2
RD3
RD4

1
A'  Ba K K  a L L  A
K '  sY
L'  nL

,
0  1

0  s 1
n0
Endogenous (4): Y , K ' , A' , L'
Predetermined (3): K , A , L
Exogenous (8): B , s , n ,  ,  , a K , a L , 
Equation RD2 is a new ideas function, which describes how the change in technology A'
depends upon the existing stocks of labor and capital, and upon the existing level of technology.
The parameter  is a measure of the effect of the existing technology level A on the change in
technology A' .
8.
Special Case: Research and Development Model When   0 and   0
SRD1 Y  A1  aL L ,
SRD2 A'  Ba L L  A
SRD3 K '  sY
SRD4 L'  nL

0  1

0  s 1
n0
Endogenous (4): Y , K ' , A' , L'
Predetermined (3): K , A , L
Exogenous (6): B , s , n , a K , a L , 
9.
Analyzing Special Case Research and Development Model When   0 and   0
Defining g A  A' / A , it follows that we can rewrite SRD2 as g A  Ba L L A 1 . Taking
the natural log and differentiating with respect to time yields

ASC1
g A'
 n    1g A
gA
Multiplying by g A yields
ASC2 g A'  ng A    1g A .
Equation ASC2 is a nonlinear differential equation in g A . We can find the steady state,
where the growth rate of technology is constant, by setting g A'  0 in ASC2. Doing so, we find
the two steady states g A  0 and
ASC3 g A 
n
.
1
If we assume there are diminishing returns to technological improvement, so 0    1 , then the
steady state growth rate of technical change given by ASC3 is positive.
Taking the derivative of ASC2 with respect to g A allows us to determine whether these
steady states are stable or unstable. We find g A' / g A  n  2  1g A . Evaluating this derivative
at the steady state g A  0 , we have g A' / g A  n  0 , which indicates the steady state g A  0 is
unstable. Evaluating the derivative at the steady state ASC3, we find g A' / g A  n  0 , so the
steady state g A  n /1    is stable. Thus, we learn that the growth rate of technology converges
over time to the rate given by ASC3.
The equation ASC3 informs us about the factors that determine the economy’s long run rate
of technological growth. A higher population growth rate n leads to a higher rate of technical
change, as does a higher level for  , which is a measure of the sensitivity of technical change to the
population level. The intuition here is that, when an economy contains more people, there is a
greater likelihood that a new idea will be produced because there are more people looking for new
ideas. As the diminishing returns to technical change gets smaller (i.e., so  approaches 1), the
steady state rate of technological improvement gets higher. This captures the idea that new
technology begets new technology.
Notice the variable a L does not appear in ASC3, which implies the portion of labor devote
to technical change does not affect the rate of technical change. This is counter-intuitive, since one
would tend to think technical change would be faster when a higher percentage of the labor force is
devoted to the production of new ideas. The reason for this result is the same reason why an
increase in the savings rate in the standard Solow Swan model does not result in a permanent
increase in the growth rate of per capita output. The reason is that temporary improvement is
eventually outweighed by diminishing returns, so that there is no long term effect. In this case, an
increase in a L would lead to a temporary increase in the growth rate of technology. However,
because there is diminishing returns to technological improvement, this increase in the growth rate
cannot be maintained. (There would have to be continual increases in the percentage of labor
allocated to producing new ideas, for the long term growth rate to be effected.)
In the long run, this economy follows a balanced growth path that mimics the paths followed
by real world developed economies. Taking the natural log of equation SRD1 and differentiating
with respect to time, we find gY  g A  g L . That is, the growth rate of output grows faster than the
growth rate of labor, and the difference between the two is the rate of technical change. The
growth rate g Y / L of per capita output is equal to the rate of technical change g A , which is what data
for the real world indicates. (Can you show this?)
10.
Analyzing General Research and Development Model
While the special case model just examined tracks the growth rate of technical change alone,
the general model tracks the growth rate for both technical change and capital.
Eliminating Y from RD3 using RD1, we obtain K '  s1  a K K  A1  a L L
by K , we can rewrite this equation as

 AL 
AGRD1 g K  c K  
K 
1
.
Dividing
1
,
where g K  K ' / K and c K  s1  a K  1  a L  . Equation AGRD1 gives the growth rate of
capital as it depends upon the level of technology, the level of labor, the level of capital, and the
model’s fixed parameters. By taking the natural log and differentiating with respect to time, we
obtain

1
g K'
 1   g A  n  g K  ,
AGRD2
gK
where g A  A' / A and n  L' / L from RD4. Condition AGRD2 describes how the growth rate of
capital changes. We will analyze AGRD2, but will wait to do so until we derive a similar condition
that describes how the growth rate of technology changes.
Dividing RD2 by A , we obtain A'  Ba K K  a L L A


AGRD3 g A  c A K  L A 1 ,
where g A  A' / A and c A  Ba K  a L  . Equation AGRD3 gives the growth rate of technology
as it depends upon the level of technology, the level of labor, the level of capital, and the model’s
fixed parameters. By taking the natural log and differentiating with respect to time, we obtain

AGRD4

g A'
 g K  n    1g A .
gA
Together, equations AGRD2 and AGRD4 are a system of two differential equations with
state variables g A and g K . We will describe the dynamics of this system using a phase diagram
analysis.
A good way to begin the phase diagram analysis is to find the “null-clines” for the state
variables g A and g K . The g K null-cline are the points in the ( g K , g A ) space where g K'  0 .
Because AGRD1 indicates g K is positive, equation AGRD2 implies
 
 
 
 
AGRD5 g   0  g K    g A  n
 
 
 
 
'
K
The g K null-cline is obtained when the equality holds in AGRD5. It separates the case where g K' ,
is positive, meaning g K is increasing, from the case where g K' is negative, meaning g K is
decreasing. Figure 1 shows these two cases
Figure 1: The g K null-cline and the dynamics of g K
gK
gK  n  g A
g K'  0
n
g K'  0
gA
The g A null-cline are the points in the ( g K , g A ) space where g A'  0 . Because AGRD2 indicates
g A is positive, equation AGRD4 implies
 
 


  n 1  
AGRD6 g A'   0  g K     
gA

 
  
 
 
The g A null-cline is obtained when the equality holds in AGRD5. It separates the case where g A' ,
is positive, meaning g A is increasing, from the case where g A' is negative, meaning g A is
decreasing. Figure 2 shows these two cases, assuming   1 .
Figure 2: The g A null-cline and the dynamics of g A
gK
g A'  0
gK  
n 1  

gA


g A'  0
n


gA
By combining the two null clines in one phase diagram, we can describe the dynamics of the
system. We do this for the case where 1   /   1 , or     1 , for this is a case where the
steady state is a sink, attracting all possible paths to the steady state ( g K , g A ) , as shown in Figure 3.
The two null clines divide the ( g K , g A ) space into four space, where the direction of movement is
different in each of the four spaces. In space I, each point ( g K , g A ) is below both null clines, which
implies g K is increasing and g A is decreasing, as shown. In space III, each point ( g K , g A ) is above
both null clines, which implies g K is decreasing and g A is increasing, as shown. Except for the
special “manifold” paths that go directly to the steady state, an initial growth rate combination
( g K , g A ) in either space I or space III will eventually enter either space II or space IV. As shown in
the figure, when a path hits a null cline there is a change in direction. Once a path enters space II,
the growth rates for g K and g A each increase and converge to the steady state ( g K , g A ) . Once a
path enters space IV, the growth rates for g K and g A each decrease and converge to the steady state
(g K , g A ) .
Figure 3: Phase Diagram for General Research and Development Case
gK
g A null cline
IV
g K null cline
III
gK
n
II
n


I
gA
gA
The basic message of the phase diagram presented as Figure 3 is, if the system is not in the
steady state, it will head to the steady state. So, it is reasonable to examine the steady state and see
what we learn. The steady state occurs when we are on both the g K and g A null clines, or where
g K'  0 and g A'  0 . When g K'  0 , we know from AGRD5 that g K  g A  n . When g A'  0 , we
know from AGRD6 that g K  n /   g A 1   /  . Using these two equations to eliminate g K
and solve for g A , we obtain
AGRD7 g A 
   n .
1     
Using AGRD7 and g K  g A  n , we find
AGRD8 g K 
1     n .
1     
Equations AGRD7 and AGRD8 give the steady state growth rates for capital and technology
as they depend upon the exogenous variables of the model. Notice the shares a L and a K , of labor
and capital allocated to research and development, do not influence the steady state growth rates
g K and g A . Nor does the savings rate s . This is somewhat surprising. Changes in these
exogenous variables do impact the growth rates in the short term. However, diminishing returns to
capital and labor remove these impacts on the growth rates over time.
A higher rate of population growth n increases the steady state growth rates g K and g A .
This is true even if   0 , which means additional labor does not influence the rate of technical
change directly. In this model, more labor generates more output, which increases the rate of
capital accumulation, and this also has a positive effect on the growth rate of technology through
research and development.
Higher values for the parameters  and  each also increase the steady state growth rates
g K and g A . These variables respectively determine the impact of labor growth and capital
accumulation on the development of technology. If  is large relative to  , then technological
improvement is coming through labor more so than through capital. Alternatively, if  is large
relative to  , then technological improvement is coming through capital more so than through labor.
This model also is consistent with the salient facts about growth experienced in more
developed nations. As long as     1 , as we have assumed, conditions AGRD7 and AGRD8
implies g K  g A . Using AGRD5, we know g K  g A  n in the steady state, which implies the
growth rate of capital is greater than the growth rate of labor, the difference being the rate of
technological improvement. This is a basic growth characteristic of more developed economies.
Finally, taking the natural log of RD1 and differentiating with respect to time, we find
gY  g K  1   g A  1   n would have to hold in the steady state. Eliminating g A from this
equation using the knowledge that g K  g A  n , we find g Y  g K , which is also characteristic of
more developed economies.
11.
LD1
LD2
LD3
LD4
Learning by Doing
Y  K  AL
1
,
A  BK 
K '  sY
L'  nL
0    1,
B  0, 0    1
0  s 1
n0
Endogenous (4): Y , K ' , A' , L'
Predetermined (3): K , A , L
Exogenous (5): B , s , n ,  , 
Equation LD2 indicates that new ideas are generated as a side effect of capital accumulation.
That is, people learn as they do. They learn how to produce better machines as they do use
machines in the production process. This better machines, or new ideas, augment the productivity
of labor through equation LD1.
12.
Analyzing the Learning by Doing Model
Substituting LD1 and LD2 into LD3, we obtain K ' / K  sB 1 L1 K   1 1 . Thus,
g K  sB1 L1 K   1 1 , where g K  K ' / K . Taking the natural log and differentiating with
respect to time, and using from LD$ the knowledge that L' / L  n we obtain
g K' / g K  1   n     1     1g K . This implies the change in the growth rate of capital is
given by
ALD1 g K'  1   ng K     1     1g K 
2
We can think of ALD1 as a new model where the variables are classified as follows:
Endogenous (1): g K'
Predetermined (1): g K
Exogenous (5): n ,  , 
Notice that the variables B and s do not appear in equation ALD1. This indicates that
neither the savings rate nor the capital productivity parameter B affects the rate at which the
growth rate of capital change. This also means they will not impact the steady state growth rate for
capital.
Setting ALD1 equal to zero, we find the steady state growth rate for capital. It is
ALD2 g K 
n
.
1
We can find the growth rates for other variables of interest by treating equations LD1 and
LD2 as auxiliary equations. Taking the natural log of LD2 and differentiating with respect to time,
we find g A  g K . Using ALD2, it therefore follows that
ALD3 g A 
n
.
1
Taking the natural log of LD1 and differentiating with respect to time, we find
gY  g K  1   g A  1   n . Using ALD2 and ALD3, it therefore follows that
ALD4 g Y 
n
.
1
Using ALD2-ALD4, we see that g Y  g K and we see gY  g A  n , and these results fit
what we observe in real world developed economics. Output and capital grow at the same rate in
the steady state. This rate is greater than the rate of population growth. And, the rate of technical
change is the differences between the growth rate of output and the population growth rate.
What is special about this learning by doing model, compared to the Solow Swan model, is
that the rate of technical change is endogenously determined by the model. It is not assumed. In
particular, equation ALS3 indicates that the steady state rate of technical change depends upon the
rate of population growth, and upon the diminishing returns parameter  . When the rate of
population growth is higher, the rate of technological improvement is greater. This is because
population growth fuels output growth, which fuels capital accumulation through saving, which
fuels technological improvement. The parameter  is the elasticity of technical change with
respect to capital accumulation. As  is closer to 1, there is less of a diminishing return effect of
capital accumulation on technical change, which magnifies the impact that population growth can
have.
13. Deriving an AK Model from a Learning by Doing Model
For the learning by doing model presented as equations LD1-LD4, assume there is no
population growth so n  0 , and assume there are no diminishing returns to learning by doing so
  1 . Then we can rewrite the learning by doing model as
LDAK1
LDAK2
LDAK3
Y  K  AL
A  BK
K '  sY
1
,
0    1,
B0
0  s 1
0    1,
Endogenous (4): Y , K ' , A'
Predetermined (3): K , A
Exogenous (5): B , s , n ,  , L
Eliminating the variable A from LDAK1 using LDAK2, we obtain Y  bK , where
1
b  BL  . Using LDAK3, we then have K '  sbK , which implies g K  sb . Taking the natural
log of LDAK2 and differentiating with respect to time, we find g A  g K . So, g A  sb must also
hold. Using the same approach the equation Y  bK implies g Y  g K . So, g Y  sb must also
hold. In summary, we know gY  g K  g A  sb . The growth rates of output, capital, and
technology are the same, and each depends upon the savings rate and the variable b .
This learning by doing presentation of the AK model allows us to offer an interpretation of the
AK model that cannot be offered with a presentation of the model that does not include the labor
variable. Capital contributes to growth both directly in production and indirectly through
improvements in technology obtained from learning by doing. Unlike the Solow Swan model and
other endogenous growth models, the savings rate affects rates of economic growth because capital
does not experience diminishing returns. Diminishing returns to capital are assumed in production,
but they are precisely compensated for by the learning by doing process. An increase in the labor
level increases the variable b in the model, which is the average product of capital. This increase
in capital productivity enhances economic growth, both directly and through the learning by doing
process.
14. Econometric Estimation of Endogenous Growth Models
Substituting LD2 into LD1 to eliminate the variable A , we obtain Y  B 
Taking the natural log we obtain
1
L1 K   1  .
EE1 ln Y   1   ln B  1   ln L     1   ln K  .
Equation EE1 is linear in the natural logs. Because of this linearity, we can estimate the
coefficients in this equation if we can obtain data on the variables Y , K , and L . If we have data
on Y , K , and L , we can construct ln Y  , ln K  , and ln L . We can then perform a linear
regression using the variable ln Y  as the dependent variable and the variables ln K  , and ln L as
independent or explanatory variables. Doing this, the regression software would give us numbers,
or estimates for the constant or intercept term 1   ln B  , the coefficient 1    on ln L , and the
coefficient    1   . Because there are three parameters here (  ,  , and B ) and three
estimates, our regression model is identified, meaning we can find the values of the parameters
from the three estimates obtained from the regression.
However, equation EE1 is not a good equation to estimate because it will not tend to have
good econometric properties. The problem is ln Y  , ln K  , and ln L will each tend to trend
upward over time. When this is the case, heteroskedasticity and serial correlation will tend to arise.
A good way to confront this kind of econometric problem is to differentiate with respect to time.
Doing so, equation EE1 becomes
EE2 gY  1   g L     1   g K .
Equation EE2 relates the growth rate of output to the growth rates of labor and capital. If we have
data on output, labor, and capital over time, then we can construct these growth rates, run the
regression associated with EE2, and find estimates for the parameters  and  . Letting t denote
a given time period, the regression equation associated with EE2 is
EE3 g Y ,t  1   g L ,t     1   g K ,t   t .
In equation EE3, the quantity g Y ,t  1   g L ,t     1   g K ,t is the model or the estimate of
the growth rate of output obtained from the learning by doing model. The quantity  t is the error
term in the regression, which tells us how far the model is from predicting the actual data during
time period t . If the econometrics properties of the model are appropriate, then the error term  t
will be randomly distributed around a mean value of zero, which implies the model sometimes
overestimates, sometimes underestimates, sometimes is off by much, sometimes by little. The
randomness of the error indicates that we do not know in advance whether the model will over
estimate or underestimate for a given period, not can we guess the magnitude of the error for a
given period.
Using data available for the U.S. from 1949 to 2004, we obtain an estimated equation
EE4 g Y ,t  1.049 g L ,t 
(0.0164)***
0.115 g K ,t ,
R 2  .7748
(0.0633)*
where the standard errors for the estimated coefficients are shown in parenthesis. The asterisks are
used to indicate the level of significance. One asterisk indicates the coefficient is significantly
different than zero at a 10 percent level of significance, two asterisks indicates significance at 5
percent, and three asterisks indicates significance at 1 percent.
Figure 4 presents the data for the U.S. for the years 1949 to 2004 for the growth rates of
output, labor, and capital used in regression EE4. The output level is measured by U.S. real gross
domestic product. The labor level is measured by the number of full time equivalent employees.
The capital level is measured by the real value of fixed private assets, multiplied by the capacity
utilization rate in manufacturing to adjust for the fact that more capital will sit idle in weak
economic times than in strong. Notice the following facts can be seen in Figure 4.
 The average growth rate for output, labor, and capital are positive. A negative growth rate is
relatively rare. (By definition, a negative output growth rate is a recession.)
 Capital growth is more volitile than output growth, and output growth is more volitile than
labor growth.
 Capital and output growth at about the same average rate, while labor grows at a slower rate.
For this data, the mean output growth rates is 3.35 percent per year, the mean capital growth
rate is 3.44 percent, and the mean labor growth rate is 1.69 percent.
Figure 5 shows how the model EE4 fits the actual output growth rate. The positive
coefficient estimates in EE4 indicate that output growth is positively correlated with both labor and
capital growth. As can be seen in Figure 5, recessions are associated with weak or negative labor
and capital growth. Our assumption in running the regression that produces EE4 is that the
variations in labor and capital growth are causing the variations observed in output growth.
(Another assumption, and one that is problematic, is that labor growth and capital growth are
independent. We would not expect independence because employers simultaneously choose how
much labor to hire and how much capital to accumulate.)
Figure 4: U.S. Growth rates for output, labor, and capital
U.S. Growth Rates: 1949-2004
Output, Labor and Capital
0.20
0.15
0.10
0.05
1945
(0.05)
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
(0.10)
(0.15)
gY
gL
gK
Figure 5: Fitting the Learning by Doing Model to the Data
U.S. Real GDP Growth Rate: 1949-2004
Actual Versus Models
0.12
0.10
0.08
0.06
0.04
0.02
1945
(0.02)
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
(0.04)
(0.06)
gY
Learning by Doing Model
The estimate 1    1.049 implies the estimate   .049 . The estimate
   1     0.115 , along with   .049 , implies  .049  1  .049  0.115 , or
  0.115  .049/1  .049  .141 . The estimate   .049 is outside the range we have specified
for our model, an indication that this model does not quite capture what is happening in the real
world. The estimate   .141 is our estimate for the elasticity of technical change with respect to
capital. This estimate indicates that a 10 percent increase in capital will produce a 1.4 percent
increase in the level of technology, an indication that technological change is responsive to capital
accumulation, but inelastically responsive.
When we re-estimate the model including a constant we obtain
EE5 g Y ,t  0.019  0.596 g L ,t  0.129 g K ,t ,
(0.002)*** (0.116)***
(0.020)***
R 2  .728
The constant estimate of 0.019 indicates that approximately 2 percent of the growth rate of output is
not accounted for by the learning by doing model. The purpose of creating the learning by doing
model is to account for technical change endogenously, and what we have learned here is that we
cannot account for all of it. We can think of the 0.019 number as independent exogenous technical
change, and put it into the model if we recast the learning by doing model letting B  e t .
Reworking the model to the econometric equation, we find the constant is an estimate of 1    ,
or 1     0.019 . With this new model, 1    .596 , so   .404 and
  0.019 /1     0.019 / .596  .032 . Thus, for this model we obtain a realistic value for the
variable  , which is the elasticity of output with respect to capital under the assumption,
abstracting from the effect capital has on technology. The number   .404 indicates that, if
capital did not affect technology, then a 10 percent increase in capital would generate a 4 percent
increase in output. The number   .032 indicates the average product of capital increases at 3.2
percent per year, for some reason that we do not understand at this point.
For this model    1     0.128 . Using   .404 , we obtain   0.46 . Unfortunately,
this is not consistent with the learning by doing hypothesis. The negative number indicates capital
accumulation has a negative impact on technological change. What we are learning here is that
econometric analysis, which forces us to relate our theory to the data, can usefully force us to
rethink our theory. The learning by doing model which endogenizes technical change in EE4
indicates capital accumulation has a negative impact on production, which does not make sense.
The learning by doing model which allows for some exogenous technical change in EE5 indicates
capital accumulation has a negative impact on technology, which indicates we unlearn by doing
rather than learn. We must continue to search for an improved model.
Let’s adjust the model by recognizing that two our assumption of constant returns to scale
with respect to labor and capital may not hold, especially since it is unlikely that our capital variable
captures the effect of all non labor inputs in production. To account for this, let’s re-specify
equation LD1 as
EE6 Y  K  AL ,

0    1,
0    1,
If production exhibits constant returns to scale, then     1 should hold, at least approximately.
While we are interested in an endogenous model of technical change, let’s start with an exogenous
model of technical change so we can first get an idea of whether a constant returns to scale
assumption is reasonable. In particular, assume
EE7 A  ae 0 1t 2t
2
 3t 3
,
which implies the level of technology changes over time at the exogenous rate
EE8 g A  1  2 2 t  33 t 2 ,
Substituting EE8 into EE7, taking the natural log, and differentiating with respect to time,
we obtain the following equation that we can estimate
EE9
g Y  1  22 t  33t 2  g K  g L .
The result of our estimation is
EE10 g Y  .0325  .0012t  .00002t 2  .149 g K  .574 g L , R 2  .7704
(.005)*** (.0004)*** (.000007)*** (.04)*** (.11)***
Thus, we find   .574 and   .149 , or     .723  1 , an indication of decreasing returns to
scale. Reconstructing EE9 using the estimates, we find g A  .057  .002t  .0003t 2 . This indicates
the rate of technical change has followed a u-shape in the U.S. over the 1949-2004 period, as shown
in Figure 7 below. This is sensible in that we might have expecting some significant technical
change after World War II in manufacturing, which would have played itself out over time. The
increase in the rate of technical change more recently might be associated with innovations in the
computer and communications industries.
Knowing that our production function is unlikely to exhibit constant returns to scale, lets
now extend the learning by doing model by replacing equation LD2 with
EE11 A  BK  (t ) L (t ) ,
where
EE12  (t )   0  1t   2 t 2  3 t 3
and
EE13  (t )   0   1t   2 t 2   3 t 3 .
This is a more general model of technical change in two respects. First, the level the technology
can be affected by both capital accumulation and labor accumulation. So, learning by doing not
only can occur because more machines are built, but also because there are more people building
machines, or more people doing. Second, the elasticity of technology with respect to capital  (t )
can change over time, as can the elasticity of technology with respect to labor  (t ) .
Substituting EE11 into the production function EE6, taking the natural log, differentiating
with respect to time, and then using EE12 and EE13, yields the following equation that we can
estimate:

 3 ln K t    g t  ln L    g t
 3 ln L t 

 2 ln L t 
g Y   0 g L     0  g K  1 g K t  ln K   2 g K t 2  2 ln K t
EE14

  g t
 3 g K t 3
3
L
3
2
1
2
L
2
L
2
Constructing the explanatory variables as needed in EE14 and running the appropriate linear
regression, we find that the variables g K t 3  3 ln K t 2 and g Lt 3  3 ln L t 2 do not have a
significant explantory effect, so we drop them. The resulting estimated equation, with all
coefficients significantly different than zero at a 10 percent level of significance, is





g Y  .436 g L  .265 g K  .0137g K t  ln K   .0002 g K t 2  2 ln K t

 .013g L t  ln L   .0002 g L t  2 ln L t
EE15
2


.
Comparing the coefficients of the variables in EE14 to those of EE15, we can see that there
are two more parameters in EE14 than estimated coefficients in EE15, which implies the regression
model is under identified. One way to still use the estimates in EE15 is to specify values for two of
the parameters in EE14. Suppose we assume   .574 and   .149 as we obtained from EE10.
This allows us to see the implications these fixed parameters have when we assume they are true
but allow for endogenous technical change. Using   .574 and   .149 , we find from EE14
and EE15 that  (t )  .764  .024t  .0004t 2 gives the time dependent elasticity of technology with
respect to capital and  (t )  .203  .023t  .0004t 2 gives the time dependent elasticity of technology
with respect to labor. These are shown in Figure 6. Our results indicate something interesting. As
the manufacturing developed over the 1949-1980 period, the evidence indicates learning by doing
came less from accumulating capital and more from accumulating labor. Learning by doing
through capital accumulation has shown a resurgence and learning by doing through labor a decline
more recently, during what we might labor the information age.
Figure 6: Elasticity of Technology with Respect to Capital and Labor
Elasticity of Technology
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
1945
1950
1955
1960
1965
1970
1975
With Respect to Capital
1980
1985
1990
1995
2000
2005
With Respect to labor
Figure 7 shows the rate of technical change estimated from the exogenous technical change
model EE10 and the endogenous technical change model EE 15. Notice the u-shape in the
exogenous technical change model matches the u-shape for the elasticity of technology with respect
to capital in the endogenous technical change model. This is an indication that changes in learning
by doing with respect to capital may have been responsible for the change in technical change we
obtain when we do not try specify from where it comes. The estimated path for technical change
for the endogenous technical change model is volitile because labor and capital growth rates vary,
and these are assumed to impact the rate of technical change. A slight u-shape is still noticable in
the endogenous technical change path, however, which is consistent with the estimate for technical
change obtained from the exogenous technical change model.
Figure 7: Estimated paths for the Rate of Technical Change for the U.S. Economy
Rate of Technical Change
0.080
0.060
0.040
0.020
1945
(0.020)
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
(0.040)
(0.060)
Exogenous Tech Change
Endogenous Tech Change
2000
2005