The Solow Growth Model
Ashantha Ranasinghe
University of Alberta
ECON 385: Intermediate Macroeconomic Theory II
1
Growth through history
2
Income gaps have widened
3
Understanding economic growth
Understanding why countries are poor and why they fail to grow is one of the most important
questions in economics
Robert Lucas: “...once you start thinking about growth and development, it is hard to think
about anything else.”
Economic growth has allowed real GDP per capita to rise by a factor of 14 between 1870 and
2007 (but not in all countries)
small differences in growth rates translate to large differences in income in the long run
4
Questions we ask
Why are some countries so much richer than others?
Why are some countries able to produce a take-off in standard of living (e.g. Botswana,
Singapore, Taiwan) while others fail to do so (e.g. Argentina, Uganda)?
What are the determinants of productivity and standard of living across nations?
Need facts to discipline our study of growth
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Disparity in GDP per worker
Ratio of GDP per worker among the richest 5 countries relative to poorest 5 countries over
time
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Investment and GDP per worker
Share of output invested in capital and GDP per worker
7
Schooling and GDP per worker
Share of output invested in education and GDP per worker
8
Population growth and GDP per worker
Strong negative correlation between population growth and GDP per worker (... as of today)
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Solow Growth Model
Central model which serves as a catalyst for understanding economic growth
Population growth is exogenous
N 0 = (1 + n)N
Production is
Y = zF (K, N ) = zK α N 1−α
Savings in economy is exogenous s ∈ (0, 1)
Economy resource constraint
C +I =Y
Evolution of capital is
K 0 = (1 − δ)K + I
0<δ<1
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Evolution of Capital
We can rewrite this as
K 0 = sY + (1 − δ)K
Converting this into effective units of labour
k0 =
1
(szf (k) + (1 − δ)k)
1+n
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Steady state
Steady state capital is where actual investment = break even investment
I
if szf (k) > (n + δ)k ⇒ k ↑ over time
I
if szf (k) < (n + δ)k ⇒ k ↓ over time
I
if szf (k) = (n + δ)k ⇒ k does not change (i.e. steady-state)
12
Equilibrium variables
Aggregate variables grow at (1 + n)
13
Effects of an increase in the savings rate
What happens to steady-state capital, output and consumption?
...but no effect on long-run growth rates
14
Effects of an increase in population growth
What happens to steady-state capital, output and consumption?
...but no effect on long-run growth rates
15
Effects of an increase in productivity
What happens to steady-state capital, output and consumption?
16
Analytical solutions for the Solow Model
Suppose Y = zK α N 1−α
output per worker is
Y
=z
N
K
N
α
= zk α
steady state capital is
α
sz (k ∗ ) = (n + δ)k ∗
1
1−α
sz
∗
k =
n+δ
steady state output is
∗
y =z
1
1−α
s
n+δ
α
1−α
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Taking the model to data
What are the implied output differences across countries based on the model
consider two countries i and j. Assume α is common. Relative output difference is
1 α 1−α
i
zi1−α nis+δ
i
yi
= 1 α 1−α
yj
sj
1−α
zj
nj +δj
yi
=
yj
zi
zj
1
1−α
si (nj + δj )
sj (ni + δi )
α
1−α
Do differences in s, n and δ account for observed cross-country output differences? We can
evaluate this against the data
18
Taking the model to data
data: Penn World Tables (PWT) 9.0 (https://www.rug.nl/ggdc/productivity/pwt/)
I restrict the sample to Canada and China in 2010
two cases: (a) standard Solow model and, (2) extended to include human capital
I
note: the productivity term z is unobserved
19
Taking the model to data
data: Penn World Tables (PWT) 9.0 (https://www.rug.nl/ggdc/productivity/pwt/)
I restrict the sample to Canada and China in 2010
two cases: (a) standard Solow model and, (2) extended to include human capital
I
note: the productivity term z is unobserved
19
Taking the model to data: Econometric Analysis
The Solow model implies for any country i
1
yi∗ = zi1−α
si
ni + δ i
α
1−α
1
α
α
ln(zi ) +
ln(si ) +
ln(ni + δi )
1−α
1−α
1−α
assuming δ is common and z’s are uncorrelated across countries, we can estimate via OLS
ln(yi ) =
ln(yi ) = β0 + β1 ln(si ) + β2 ln(ni + δ) + εi
if theory is correct we should expect that β1 > 0 and β2 < 0
This is what we find BUT differences in s and n explain only about 20 percent of output
differences
I
when human capital (educational differences) is added, the model can account for close 60
percent of observed output differences... but there are econometric issues related to robustness
20
Taking the model to data: Econometric Analysis
data using the PWT 9.0 sample
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Taking the model to data: Econometric Analysis
data using the PWT 9.0 sample
22
Solow Residual and Growth Rates
Previous analysis implied much of output differences is driven by differences in z
Y = zK α N 1−α . If α = 0.3 then implied productivity is
ẑ =
Ŷ
K̂ α N̂ 1−α
We have data on Ŷ , K̂ and N̂ for hundreds of countries over the last 50 plus years (see for
example, Penn World Tables)
Which means we can back out ẑ for each country and evaluate differences in ẑ across countries
and across time
23
Data and Calculating Annual Growth rates
Annual growth rate is
1
Xn n−m
−1
Xm
we do this because data is not on an annual basis
1
Y1971 10
g1961,1971 =
−1
Y1961
gm,n =
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Data and Calculating Annual Growth rates
Growth miracles?
however, many of these countries grew due to investment (high s) and not due to z
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