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Instructor’s Manual
Chapter 16: Endogenous Growth Theory
Chapter 16: Endogenous Growth Theory
Problem 1
In the neoclassical model, only technological progress, which is exogenous and not explained
within the model, can generate sustained economic growth. In endogenous growth models,
technology growth is once again the source of growth, but it is explained endogenously within the
model. In endogenous growth models, changes in exogenous variables that increase the capital
stock, such as a higher savings rate, can increase productivity, which, as a result, can change the
growth rate within an economy.
Problem 2
Perfect capital mobility exists when the world capital market is open and loanable funds can
move freely in and out. One counterfactual implication of perfect capital mobility in neoclassical
growth models is that there should not necessarily be a close relationship between domestic
savings and investment. In the real world, the level of domestic investment is closely related to
the level of domestic savings across most countries. Secondly, perfect capital mobility predicts
that real interest rates should be equalized across countries, implying that marginal products of
capital, capital to labor ratios, and GDP per person also should be equalized across countries as
well. Obviously, this is not consistent with what we observe.
Problem 3
In the neoclassical model, higher investment rates increase the level of per capita GDP within a
country. However, Panel B in Box 16.1 illustrates that even after controlling for differences in
population growth rates, there does not appear to be a strong positive relationship between
investment rates and GDP per capita. This is very much at odds with the neoclassical model.
Problem 4
In the neoclassical model, changes in the investment rate change the level of per capita GDP but
do not affect the growth rate of GDP in the long run, which is determined solely by the
exogenous rate of technology growth. Figure 16.8 illustrates that there is a positive correlation
between investment and GDP growth across countries, contrary to the predictions of the
neoclassical growth model. This is one of the primary reasons for the development of
endogenous growth theory, which attempts to explain how investment can affect long run growth
rates in an economy.
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Chapter 16: Endogenous Growth Theory
Problem 5
Endogenous growth models, such as the learning-by-doing model, predict that higher investment
rates can increase growth rates. The reason is that investment does not suffer from diminishing
returns to scale at the aggregate level because of the positive productivity externalities associated
with higher levels of private capital. As a result, endogenous growth models can explain the fact
that higher investment rates are positively correlated with higher growth rates, as seen in Figure
16.8.
Problem 6
An externality, as the term suggests, is an effect that is external to an economic agent's decision
making, and for which she does not have to pay directly. For instance, the education level of the
workforce is an externality for an individual producer. While a more educated workforce will do
the same work more efficiently, the producer does not have to pay the workers for their education
level.
One implication of the theory of endogenous growth is that the social production function is
linear in capital, i.e., the social production function does not have the property of diminishing
marginal product of capital. But the private production function does display a diminishing
marginal product and this property is important since it allows us to assume that the capital
elasticity and the labor elasticity of the private production function add up to 1 (constant returns
to scale). If the private production function was linear in capital then it would display a constant
marginal product of capital. This in turn would mean that the capital elasticity of the private
production function was 1 and the neoclassical theory of distribution would break down since if
factors were paid their marginal products, factor shares of national income would add up to more
than 1.
In endogenous growth theory, capital and labor are paid their private marginal products. But
social marginal products exceed private marginal products because of the social externality of
knowledge accumulation. Each producer takes the knowledge function, Q, as given. An
individual firm is too small for its own investment decision to influence the aggregate capital
stock. Hence the firm will assume that Q is exogenous. This assumption implies that each
producer's production function still displays diminishing marginal returns to capital, but the social
production function (constructed by explicitly incorporating the externality, see equation 14-13 in
the text) is linear in capital.
Problem 7
In the theory of learning by doing, the growth equation is
K t 1  (1    sA) K t .
Therefore, the parameters influencing the rate of growth are s, A, and .
 A higher s increases the amount of saving, and hence total investment. As a result, the capital
stock grows. Since capital exerts an externality on the production function through the
knowledge function, this means that output will be still higher1. Thus the growth rate of the
economy will be higher too.
1
In the neoclassical model this is true as well, but after a point any increase in K will depress the marginal
product of capital so much that investors will not want to invest any more. Not so, in the endogenous
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Chapter 16: Endogenous Growth Theory


A higher A improves the productivity of capital -- the same capital is able to produce more
output, more saving, and hence more capital in the future. Consequently the growth rate will
be higher.
A higher  means more resources have to be devoted to maintaining the existing capital stock
rather than being used to build new machines and factories. Hence the growth rate will be
lower.
Problem 8
a.
If s = .1,  = .05, A = L = 1, then the steady state level of the capital stock in the
neoclassical model without technology is k  ( sA /  )1 /(1 ) = 4. This economy will
experience a steady state growth rate of zero because technology is fixed.
b.
If Q = K then the social production function can be rewritten as Y = AK. In this
endogenous growth model, the growth equation is K t 1  (1    sA) K t  1.05K t , so
output is growing at 5% a year. No steady state level of capital exists within this model
because capital is constantly growing.
c.
In the model with endogenous technology (Q = K), diminishing returns to scale do not
exist in the social production function. As a result, sustained increases in per capita
growth are possible and are generated by the productivity externalities that result from
increases in the private capital stock.
Problem 9
Individual firms face constant returns to scale in production and, as a result, diminishing marginal
returns when increasing their private capital stock. However, increases in private capital stocks
create positive productivity externalities at the aggregate level. As a result, the social production
function exhibits increasing returns to scale, meaning that doubling capital and labor will more
than double aggregate output. These externalities also mean that diminishing marginal returns do
not exist in the social production function so that continuous per capita GDP growth is possible
without exogenous increases in technology.
Problem 10
Since A = 1,  = 0.1 and s = 0.2, the growth equation is K t 1  1.1K t , from which we can obtain
the growth rate of capital as
K t 1
 1  0.1 . Therefore, capital grows at the rate 10% per year.
Kt
growth model. Current investment creates more incentives for future investment as the rate of return on
capital is constant and does not ever fall.
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Chapter 16: Endogenous Growth Theory
Problem 11
a.
The steady-state per capita capital stock in the neoclassical growth model is
1
 sA  1
k   .
 
Hence, steady-state per capita GDP, using the per capita production function, is

 sA  1
y  A  .
 
Using this formula, and a value  = 1/3, as the neoclassical model does, the third column
in the table below calculates the per capita steady states for each of the five countries. The
fourth column expresses steady-state per capita GDP as a fraction of U.S. GDP per person.
Country
The United States
The Philippines
Mexico
Japan
Ghana
b.
Saving rate
0.16
0.17
0.18
0.25
0.09
Steady state y
1.26
1.30
1.34
1.58
0.95
Fraction of U.S. y
1.00
1.03
1.06
1.25
0.75
As in Problem 3, the growth rate of the capital stock is given by
K t 1
 1  sA   .
Kt
Hence the growth rate of per capita GDP is also the same, i.e., sA - . The second column
of the table below computes this predicted growth rate, and the third column expresses
the growth rate of each country relative to the U.S. growth rate.
Country
The United States
The Philippines
Mexico
Japan
Ghana
Growth rate
6%
7%
8%
15%
-1%
Fraction of U.S.
growth rate
1.00
1.16
1.33
2.50
-0.17
Problem 12
According to the endogenous growth theory, the production function is linear in capital K:
Y = AK, where A is a constant. The marginal product of capital is simply A, which is independent
of the saving rate. In other words, if we believe that all countries have access to the same
technology, Y = AK, then they all have the same marginal product of capital, A.
If capital is freely mobile internationally, it will flow in response to the relative marginal
product of capital. Countries with higher marginal returns on capital will attract investment from
abroad. But here, since all countries have the same marginal product, there is no incentive for
capital to flow from one country to another.
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Problem 13
In the neoclassical growth model, diminishing returns to scale imply that the marginal product of
capital is higher in a country with a lower capital to labor ratio, everything else being equal. As a
result, the marginal product should be higher in country A, the country with the lower initial
capital level. However, in endogenous growth models, social productivity varies with the
aggregate capital stock. Thus, the marginal product of capital will be larger in the country with
the larger capital stock. In this case, country B will have the higher marginal product of capital,
contrary to the prediction of the neoclassical growth model.
Problem 14
a.
Divide both sides of the production function by labor, N, and then use the fact that
N  N  
1/ 
:
Y aK   (1  a ) N  

N
N
1/ 
  K 

 a   (1  a )
  N 

1/ 
.

Therefore, the per capita production function is simply: y  ak   (1  a )
b.

1/ 
The marginal products of capital and labor are obtained by taking the derivatives of the
original production function with respect to capital and labor. Therefore,
1 
MPK =
1 /  1
Y
1
Y 
 aK   (1  a ) N  
.  aK  1  aY 1  K  1  a  
K 
K
MPN =
1 /  1
Y 1
Y 
 aK   (1  a ) N   .  (1  a ) N  1  (1  a )Y 1  N  1  (1  a ) 
N 
N
,
1 

where we have made use of the fact that aK   (1  a ) N 
c.

1 /  1
 Y  
1 /  1
 Y 1  .
Labor's income is wage income, N. The real wage rate , is equal to the marginal
product of labor if labor markets are perfectly competitive. Hence, labor's share of
income is:
N
Y
(1  a )Y / N 
Y
1 

.N


(1  a )Y 1  N 
Y 
 (1  a )  .
Y
N
Since this depends upon the amount of labor hired and output produced, it is not a
constant in general. However notice that if  = 0, then labor's share of output becomes a
constant equal to (1-a), which is the Cobb-Douglas case.
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