GrowthRatesandLogarithms: ARefresher
Growth Rates and Logarithms: A Refresher
Prof. Lutz Hendricks
Econ520
January 5, 2016
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U.S. Economic Growth
It looks like U.S. growth has been accelerating? Is that true?
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What Is a Growth Rate?
We need to understand the math of growth rates. The growth rate g is defined as
g = x (t + 1) - x (t)
(1)
x (t)
Or:
x (t + 1) = (1 + g) x (t)
(2)
Example: x (t) = 100, g = 5%. Then x (t + 1) = 1.05 ? 100.
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Growth rates over multiple periods
If we take multiple periods: x (t + n) = (1 + g)n x (t)
Example: GDP per capita grows at 1.8% per year. y2002 = 30, 000 y2003 = 1.018 ? $30, 000 = $30, 540. y2003 = 1.018y2002 = 1.0182 y2001.
Example: In 50 years, y grows by 1.01850 = 2.44.
(3)
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Calculating the average growth rate
The average growth rate answers the question:
Which constant growth rate would change yt to yt+n in n years?
Start from
yt+n = yt ? (1 + g)n
(4)
and solve for g.
(1 + g)n = yt+n/yt
(5)
1 + g = (yt+n/yt)1/n
(6)
Example: Average GDP growth since 1870. The annual growth rate is calculated from y2000 = y1870 (1 + g)130. Therefore: g = (y2000/y1870)1/130 - 1 = 0.0179 = 1.79% p.a.
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