Long Run Growth

 In this chapter, we learn:
some facts related to economic growth that later chapters will seek to explain.
how economic growth has dramatically improved welfare around the world.
that this growth is actually a relatively recent phenomenon.
some tools used to study economic growth, including how to calculate growth
rates and why a “ratio scale” makes plots of per capita GDP easier to understand.
 The United States of a century ago could be mistaken for Kenya or Bangladesh today.
 Some countries have seen rapid economic growth and improvements to health
quality, but many others have not.
 We will trace income and health trends as reported in the UNHDR
 Sustained increases in standards of living are a recent phenomenon.
 Modern economic growth only emerged in the most recent two or three centuries.
 Sustained economic growth emerges in different places at different times.
 Per capita GDP – living standards – differ remarkably around the world.
 The Great Divergence is the era of increased difference in standards of living across
 Before 1700, nations differed only by a factor of two or three, while today it is over a
factor of 50 for several countries.
 From 1870 to 2000,
United States per capita
GDP rose by nearly 15fold.
 A typical college student
today will earn a lifetime
income about twice his
or her parents.
 If we observe a constant
doubling time for a
variable, e.g., population
or real per capita income,
the variable is growing at
a constant annual rate.
 If income is growing at a constant rate it will be a straight line on a ratio scale; but if
growth rates are rising, the slope will be increasing.
 When plotted on a ratio scale, it is easy to see that per capita GDP in the United States
has grown at approximately 2 percent per year over the last 130 years.
The Rule of 70, Doubling Time, and the Ratio Scale
Time to double (T) at constant growth rate (g):
2 = (1 + g)T
Ln 2 = 0.693 ≈ 0.70 = T Ln (1 + g) ≈ T g
since for small g, Ln(1 + g) ≈ g
T = 70/g
where g is expressed in percent
 The Rule of 70 says that 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.
 Seemingly small differences in growth rates result in large differences over time.
 The time it takes to double only depends on the growth rate and not the initial value.
 If we know a variable’s doubling time, T, its average growth rate is
g = 70/T
The Definition of Economic (or any) Growth
 A growth rate, g, is the percentage change in a variable.
gx = dx/x
 From what we know about calculus,
d(x * z) = dx + dz
d(xn) = d(x * x * x…* x) = n dx
d[Ln(x)] = dx/x = gx = (xt+1 – xt)/xt
 The value of variable x next year is then
xt+1 = xt * (1 + gx)
After t years of constant growth at rate g, a variable whose initial value is x0 will equal
xt = x0 (1 + g) (1 + g) … (1 + g) = x0 (1 + g) t
If we know xt and x0 , the average compound rate of growth over t years is
g = (xt / x0 )1/t – 1
 Division of two variables implies subtraction of the growth rates.
z = x/y = x1 y-1
dz = y-1 d(x1) + x1 d(y-1 ) = y-1 dx – 1y-2 x1 d(y)
dz/z = (y-1dx)/(x1 y-1) – (1y-2 x1 d(y))/(x1 y-1) = dx/x – y-1 dy = dx/x – dy/y = gx – gy
 Multiplication of two variables implies addition of the growth rates.
z = xy = x1 y1
dz = y1 d(x1) + x1 d(y1) = y1 dx + x1 d(y)
dz/z = (y1 dx )/(x1 y1) + (x1 dy)/(x1 y1) = dx/x + dy/y = gx + gy
 When a variable is raised to an exponent, the growth rate of the quantity is the
exponent times the growth rate of the variable.
z = xa = x x … x so dz = a xa-1 dx = and dz/z = (a xa-1 dx )/ xa = a dx/x = a gx
Y = Real GDP = Population * Per Capita Real GDP
gY = g N + gy
gy = gY - gN
Modern Growth around the World
 After World War II,
growth in Germany and
Japan accelerated.
 Convergence is the idea
that poorer countries will
grow faster to catch up to
the level of income in
richer countries.
 Brazil had accelerated
growth until 1980 and then
stagnated, while China and
India have had the reverse
 When comparing levels of
income on a ratio scale,
recall that data points that
are half below another
country are actually much
lower because the numbers
on the vertical axis double.
 Some countries have
exhibited a negative
growth rate over a forty
year period.
 Other countries have
sustained nearly 6 percent
growth over the same
 Most countries have
sustained about 2 percent
 Small differences in
growth rates result in large
differences in standards of
 The bulk of the world’s
population is substantially
richer today than it was in
 The fraction of people living
in poverty has fallen since
 A major reason for these
changes is the recent
economic growth in China
and India – which together
account for 40 percent of the
world population.
Growth Rules For Cobb-Douglas Production Function
Yt = AtKt1/3Lt2/3
 Applying rules of growth rates will show that
g(Yt) = g(At) + (1/3)*g(Kt) + (2/3)*g(Lt)
We can estimate total factor productivity growth, g(At), as a residual
g(At) = g(Yt) - (1/3)*g(Kt) - (2/3)*g(Lt)
The Costs of Economic Growth
 The benefits of economic growth include: improvements in
health, higher incomes, and an increase in the range of
goods and services, among other things.
 Costs of economic growth include environmental problems,
global warming, income inequality across and within
countries, and a loss of certain types of jobs.
 Economists generally have a consensus that the benefits of
economic growth outweigh the costs.
3.7 A Long-Run Roadmap
 Are there certain policies that will allow a country to
grow faster?
 If not, what about a country’s “nature” makes it grow
at a slower rate?
 Viewed over the long course of history, sustained growth in
standards of living is a very recent phenomenon. If the
130,000 years of human history were warped and collapsed
into a single year, modern economic growth would have
begun only at sunrise on the last day of the year.
 Modern economic growth has taken hold in different places
at different times. Since several hundred years ago, when
standards of living across countries varied by no more than
a factor of 2 or 3, there has been a “Great Divergence.”
Standards of living across countries today vary by more
than a factor of 60.
CHAPTER 3 An Overview of Long-Run Economic Growth
 Incomes in the poorest countries of the world are
probably no more than twice as high as average
incomes around the world a thousand years ago.
 Since 1870, growth in per capita GDP has averaged
about 2 percent per year in the United States. Per
capita GDP has risen from about $2,500 in 1870 to
more than $37,000 today.
 Growth rates throughout the world since 1960 show
substantial variation, ranging from negative growth
in many poor countries to rates as high as 6 percent
per year in several newly industrializing countries,
most of which are in Asia.
 Growth rates typically change over time. In
Germany and Japan, growth picked up considerably
after World War II as incomes in these countries
converged to levels in the United Kingdom. Growth
rates have slowed down as this convergence
occurred. Brazil exhibited rapid growth in the 1950s
and 1960s and slow growth in the 1980s and 1990s.
China showed the opposite pattern.
 Economic growth, especially in India and China, has
dramatically reduced poverty in the world. In 1960,
2 out of 3 people in the world lived on less than $5
per day (in today’s prices). By 2000, this number
had fallen to only 1 in 10.
Real Per Capita Incomes in Constant Dollars
Real Per Capita Incomes in Constant Dollars

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