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Chapter 3-4
EXTENSIONS:
Endogenous
Investment and
Population
Growth
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Extensions to the basic Solow
model
We will pursue two of them here
• Endogenous investment and saving
• Population growth
We will do more (human capital, technology,
efficiency) in the next classes
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Extensions to the basic Solow
model: endogenous investment
Did we get an answer to our question of what determines
income or growth gaps across countries from the basic
Solow model?
Yes, but only a partial one
• We do not know – in turn - why investment rates differ
1. Is it because of differences in saving rates?
2. Or is it because foreign capital inflows are so substantial
that the extent of domestic saving ends up being irrelevant
for domestic investment?
– We will answer question 2 later on (speaking of
globalization); for now we tackle question 1
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The saving rate is not constant
across income groups by decile of
income per capita
This suggests that the true causation between saving
and per-capita Gdp may go the other way around: from
Gdp to saving, and not from saving (and I) to Gdp
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Implications of Solow with
endogenous invesment
• The convergence property may break down
• Not necessarily true anymore that poorer
countries will grow faster than richer countries
– Why? See it graphically yourself
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Solow Model with Saving Dependent
on Income Level: result, multiple
steady states & poverty traps
When we allow for saving rates to differ across countries
(with rich people saving more than poor people), the Solow
model becomes a model with two steady state equilibria
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Extensions to the basic Solow
model: population growth
So far: population constant
What if population is not constant? What is the
relation between population growth and income?
What is the relation between population growth and
(transitional) growth?
• Fortunately, we can adapt the Solow model and
answer these questions in a relatively
straightforward way
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Starting point: a negative relation
between population growth and income.
How do we make sense of this?
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
4-8
Idea: amend the Solow model
Main point: with population growth rate, per-capita
capital accumulation gets “diluted”
Suppose that population growth rate = n (say 1% per
year)
To keep capital per worker constant, we need to
supply the new workers around with the same
amount of capital as the old workers
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How the maths change (in a
Cobb-Douglas world)
The equation of k accumulation changes as follows:
Δk = γAkα – δk – nk = γAkα - (δ+n) k
The steady state equilibrium equation becomes:
γAkα = (δ+n) k
And the equation for transitional growth becomes:
Δk/k = γAkα-1 - (δ+n)
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Implications
Once we allow for population growth, the following
happens:
• for given investment and savings, capital accumulation
per worker will be lower as well
• Gdp per worker will be lower as well
• Transitional growth of Gdp will be accordingly lower
In other words, population growth is associated with:
• Lower steady state level of Gdp and capital per worker
• Lower growth along the transition
Here is why we say that population growth in the Solow
model leads to “dilution”
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Graphically -- The Solow Model
Incorporating Population Growth
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
4-12
Quantitative analysis with the Solow
model and population growth
kss =[γA/(n+δ)]1/(1-α)
yss =A(kss)α = A1/(1-α)[γA/(n+δ)] α/(1-α)
Which is the numerical effect of “n” on yss and kss in practice?
To make things simpler, we compare yss for two countries (say “Oecd”
and “Asia”) with same A, γ and δ but different “n”
• yssOecd/y ssAsia = [(nAsia+δ)/(nOecd+δ)] α/(1-α)
– Take δ=5% and α=1/3. Take nOecd=0% and nAsia=4%
• yssOecd/y ssAsia = [(0.04+0.05)/(0.00+0.05)] ½ =1.34
• An Oecd country, given that its population growth is much lower than
in an Asian country, will enjoy higher Gdp per worker by some 34%
• This compares with usually much larger actual differences
(US/India=1/0.13, which is about 8:1)
This means that differences in population growth only partially account
for steady state income gaps
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The Solow model with different
population growth and investment rates
We can improve the predictive ability of the Solow
model by simultaneously allowing for different
investment and different population growth rates
AT THE SAME TIME
Things become algebraically more complicated (but
conceptually NOT!). For each country:
yiss =A(kiss)α = A1/(1-α)[γiA/(ni+δ)] α/(1-α)
• Then take a value for A, δ and α and redo the same
exercise as before (you do it yourself)
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