Macroeconomics
Overview of Long-Run Growth
Lecture 4
Nausheen Sodhi
Monsoon 2026-27
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Very long run vs
Long run
https://web.archive.org/web/20211102093357/http%3A%2F%
2Fwww.ggdc.net%2Fmaddison%2Foriindex.htm
Growth over the very
long run
• Sustained increases in standards of living
are a recent phenomenon
• The Great Divergence: The increased
difference in standards of living across
countries started in the late 1700s
• Before 1700: Per capita GDP in nations
differed only by a factor of two or three
• Today: Per capita GDP differs by a factor
of 50 for several countries.
Rethink ‘time’ on the X-axis as music beats
Per Capita GDP of India since 1960
Per capita income in real terms has become almost 5 times in last 50 years, 3.5 times in last 30 years.
A typical Indian earns 3.5 times than his parents did 30 years ago
Incomes are rising with ever increasing amount every year.
Definition of Economic Growth
Economic Growth : Growth in per capita GDP = Percentage change in GDP per capita between
period t+1 and t
𝑦𝑡+1 − 𝑦𝑡
𝑔ҧ =
𝑦𝑡
or
𝑦𝑡+1 = 𝑦𝑡 (1+ 𝑔)ҧ
Constant Growth Rule
If a variable starts at some initial value y0 at time 0 and grows at a
constant rate 𝑔ҧ , the value of the variable at future time t is given by:
ഥ )𝒕
𝒚𝒕 = 𝒚𝟎 (𝟏 + 𝒈
• If initial population is 6 billion, and it grows at 2% every years, what
will be the population in 100 years?
• With 𝐿0 = 6 𝑏𝑖𝑙𝑙𝑖𝑜𝑛 and 𝑛ത = 0.02 , 𝐿100 = ?
The Rule of 70
• If 𝑦𝑡 grows at a rate of g percent per year, then the number of years it
takes 𝑦𝑡 to double is approximately equal to 70/g. The time it takes to
double only depends on g and not the initial value of y
• Small differences in growth rates result in large differences over time
• If a country’s income grows at 1% p.a., then it takes about 70 years
for income to double. but, if growth is slightly faster at 5% p.a., then
income doubles every 14 (70/5) years
How long does a country take to double its
incomes?
If income starts at 𝑦0 , how many years it takes until 𝑦𝑡 = 2𝑦0 ?
That is, 𝑦𝑡 = 2𝑦0 = 𝑦0 (1 + 𝑔)ҧ 𝑡 , or
2 = (1 + 𝑔)ҧ 𝑡
Taking logs on both sides:
ln 2 = 𝑡 × 𝑙𝑛(1 + 𝑔)ҧ
Note: ln 2 = 0.7 and 𝑙𝑛 1 + 𝑔ҧ ≈ 𝑔ҧ
Thus, 𝑡 = 0.7/𝑔ҧ
Shortcomings of standard scale
From the figure on slide 10, can you tell if the growth
rate of population is constant over the years?
Not really!
The ratio scale makes it easier to see what’s happening
to the growth rate
Standard Scale vs
Ratio Scale
• Ratio Scale: a plot where equally spaced
tick marks on the vertical axis are labeled
consecutively with numbers that exhibit a
constant ratio, like “1, 2, 4, 8, . . .” (a
constant ratio of 2) or “10, 100, 1,000,
10,000, . . .” (a constant ratio of 10).
• Plotted on a ratio scale, a variable
growing at a constant rate appears as a
straight line.
US GDP – standard scale vs ratio scale
Can you tell 𝑔ҧ from the ratio scale graph?
• Check the number of years it takes for incomes to double to compute
growth rate.
• What growth rate can you compute from the same?
• Solving for 𝑦𝑡 = 𝑦0 (1 + 𝑔)ҧ 𝑡 , we get:
1/𝑡
𝑦𝑡
𝑔ҧ =
−1
𝑦0
• This can be applied even data does not exhibit the constant growth.
• In which case, we get an average annual growth rate
Quick recap
• Suppose U.S. per capita GDP was $3600 in
1870 and $61,000 in 2018. What is the
average annual growth rate?
Modern Economic Growth around the World
• In late nineteenth century, United Kingdom was
the richest country, but United States grew faster
and hence became the richest country
• Convergence: Poorer countries will grow faster
to “catch up” to the level of income in richer
countries
• After World War II, growth in Germany and
Japan accelerated to converge to the level of UK
• China’s growth has sharply accelerated post
1978, India’s growth rate has sharply post the
1990s
• But Argentina diverged: it had accelerated
growth until 1980, post which it slowed down
considerably
https://youtu.be/0Ca5W7NcOWQ?si=J29z3Ke0dy6Ob2lx
Broader
sample of
countries
Some Useful Properties
of Growth rates
If x and y have growth rates of 𝑔𝑥 and 𝑔𝑦 then the
following rules apply (if 𝑔𝑥 and 𝑔𝑦 are small):
1.
If z = x/y, then 𝑔𝑧 = 𝑔𝑥 - 𝑔𝑦
2.
If z = x * y, then 𝑔𝑧 = 𝑔𝑥 + 𝑔𝑦
3.
If z = 𝑥 𝑎 , then 𝑔𝑧 = 𝑎 ∗ 𝑔𝑥
Examples of z
Growth Rates as log
differences
𝑦𝑡+1 = 𝑦𝑡 1 + 𝑔
Taking log on both sides,
ln (𝑦𝑡+1 ) = l𝑛(𝑦𝑡 ) + 𝑙𝑛 1 + 𝑔
For small rates of growth,
𝑙𝑛 1 + 𝑔 ≈ 𝑔
Thus,
g = ln(𝑦𝑡+1 ) − ln(𝑦𝑡 )
Using log differences to compute growth rates
Example of z= xy
Benefits:
Higher incomes
Benefits and
Costs of
Economic
Growth
Improved health, higher life expectancy, reduced
infant mortality
Increase in variety of goods and services
Costs:
Environmental problems
Income Inequalities
Loss of certain types of jobs