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Growth rates
“Hats”1 are growth rates, or percentage changes, in any variable. Take for
example Y , the GDP in year t compared the year before, t − 1. We have:
Ŷ =
Example 1
calculate Ŷ .
∆Y
Yt − Yt−1
Yt
=
=
−1
Y
Yt−1
Yt−1
if Y or GDP grows from 100 to 110 between 2000 and 2001,
110 − 100
Yt − Yt−1
=
= 0.1
Yt−1
100
Note that it is always possible to write:
Yt = Yt−1 + Ŷ (Yt−1 )
which is more commonly expressed as the equivalent.
Yt = Yt−1 (1 + Ŷ )
(1)
This says that last year’s GDP times “one plus the growth rate of GDP” is this
year’s GDP.
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Hat calculus
Rule 1. Product Rule for multiplication of two variables. Take two variables Y
and Z such that
X =YZ
The product rule is:
X̂ = Ŷ + Ẑ
In words: When X is equal to the product of Y and Z, the growth rate of X
is the growth rate of Y plus the growth rate of Z.
Example 2 Real and Nominal GDP. The price level is measured by the GDP
deflator. This is an index which starts at 100 for some arbitrarily defined base
year. The GDP deflator is a measure of the average prices of goods and services
in the economy. The inflation rate is the percentage increase in the price level
from one year to the next. The consumer price index (CPI) is another measure
of inflation that does not take into account the whole of GDP, but only those
goods purchased by consumers (leaves out investment goods for example). The
CPI is the consumer price index with weights from the market basket, but the
investment and government expenditure price indices use market baskets appropriate to their concept. The producer price index (PPI) is an average of the
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Sombreros en español.
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prices received by producers of goods and services at all stages of the production
process. It is the wholesale price index. The GDP deflator is defined as
P = Yn /Ybase
where Yn is nominal GDP and Ybase is the same GDP but measured in base year
prices. This can also be expressed as
Ybase = Yn /P
so that the P “deflates” any rise in nominal GDP due to inflation. Finally, we
can also write
P Ybase = Yn
the GDP deflator times real GDP gives nominal. Now take hats
P̂ + Ŷbase = Yˆn
This says that the inflation rate, P̂ , plus the growth rate in real GDP is the
growth rate in nominal GDP. Thus, if we knew that nominal GDP was growing
at 5.5 percent but inflation was 3.2 percent, then real GDP would only be growing
at 5.5 − 3.2 = 2.3 percent.
Example 3 The quantity equation is
M v = P Yn
where M is the money supply, v is velocity. Velocity depends on how fast the
money supply circulates through the economy. If the velocity is fast then a
small amount of money can support a large GDP and vice-versa. Velocity is
determined by the institutional arrangement of society. For example with the
rise in the use of credit cards, velocity of money rises since less actual money
is required for the same GDP. Applying hats to this example:
M̂ + v̂ = P̂ + Yˆn
Normally, the institutional framework of society does not change that quickly, so
we can assume that v̂ = 0. Then if the money supply is growing by 10 percent,
but real GDP is growing only by 4 percent, we know that inflation, by the hat
equation, must be 6 percent.
Rule 2. Quotient rule for when the two variable are divided
X = Y /Z
The quotient rule is:
X̂ = Ŷ − Ẑ
In words: When X is equal to the quotient of Y and Z, the growth rate of
X is the growth rate of Y minus the growth rate of Z. Note that this is just
rearranging the product rule.
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Example 4 Define GDP per capita, y, as
y=
Y
N
where N is the population of the whole country (as distinct from the labor force,
L ≤ N ) and Y is GDP of the whole country. To calculate the growth rate of
income per capita, write:
ŷ = Ŷ − N̂
Rule 3. Exponent rule
Y = Xz
where z is a constant, then:
Ŷ = z X̂
Example 5 Let the labor supply be related to output with the following function:
Y = L0.5
This gives:2
Ŷ = 0.5L̂
Can you combine the two? i.e., how would we express:
Y = aX b
in terms of growth rates or hats? Answer:
Ŷ = â + bX̂
but if a is a constant, then â = 0.
Rule 4. Now there is a special case when X is e, the base of the natural
logarithm system:
Y = egt
Ŷ = g
Example 6 Over the twentieth century an economy has grown on average about
2.5 percent in real terms. Express this in terms of e, the base of the natural log
system. Answer: Y = e0.025t
2 Proof:
calculate dots:
Ẏ = 0.5L(0.5−1) L̇
divide by level to get percentage change:
Ẏ /Y = 0.5L−0.5 L̇/L0.5
0.5L−0.5 L̇/L0.5 simplify:
Ẏ /Y = Ŷ = 0.5L̇/L = 0.5L̂
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Note that like all applications of calculus, these rules only apply for “small”
changes; for large changes they are only approximations. The rule of thumb:
don’t apply hats to growth rates over 10 percent.
Rule 5. There is no rule for addition. The best we can say is that when
we have a sum
X =Y +Z
then if both Y and Z are growing at the some common rate, then X will also
be growing at that same rate. Otherwise we have
∆X
∆X
X
∆X
X
X̂
=
=
=
=
∆Y + ∆Z
∆Y + ∆Z
X
∆Y Y
∆Z Z
+
Y X
Z X
wŶ + (1 − w)Ẑ
where w = Y /X and so (1 − w) = Z/X. This is not that useful since as Y and
Z grow, w changes. In the special case of Ŷ = Ẑ, we have X growing at the
same rate, but that is about it.
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Constant versus variable growth rates
Thus the path for any variable can be represented two ways: (1) as a function
of time as expressed in the table above; (2) as a starting point with a growth
rate. If as in table 1, the growth rate is variable, then we must write:
Yt = Yt−1 (1 + Ŷi )
where the subscript on Ŷ indicates that it can change from period to period. If
we impose a constant growth rate, we have equation 1 above.
If the growth rate of a variable X is g then we can write:
X1 = (1 + g)X0
that is the level of X in period 1 is just whatever it was in period 0 plus gX0 ,
the growth from period 0 to period 1. The same can be said for period 2:
X2 = (1 + g)X1
Now we can combine these two equations to write
X2
=
(1 + g)(1 + g)X0
X2
=
(1 + g)2 X0
and now generalizing
Xt = (1 + g)t X0
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(2)
Table 1: Macroeconomic data
Year
CPI
Percent Change
Inflation Rate
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
100
101
102
111
121
122
123
126
131
138
148
–
0.01
0.01
0.09
0.09
0.01
0.01
0.02
0.05
0.05
0.07
0.01
0.01
0.09
0.09
0.01
0.01
0.02
0.05
0.05
0.07
Source: Made up data
To get an average rate of growth, g, we can write
Xt
= (1 + g)t
X0
and taking the tth root
Xt
X0
1/t
=1+g
and now solving for g
g=
Xt
X0
1/t
−1
(3)
This gives the average rate of growth g for t growth periods knowing the beginning level and the ending level.
Example 7 Consider the path in table 1 for the consumer price index, what is
the average inflation rate? Answer: Using equation 3
g=
148
100
1/10
− 1 = 0.039983
or 4 percent.
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Doubling Times
The average growth rate equation gives rise to a convenient expression for doubling time, often called the “rule of 72”. Here is how it works. The basic
equation for a growth process at a constant rate is given by equation 2 above
Xt = (1 + g)t X0
Doubling time means how much time would it take for X0 to double? That is
for
Xt /X0 = 2
For a given growth rate, we need only solve
2 = (1 + g)t
Taking the natural log of both sides
ln 2 = 0.693 15 = t ln(1 + g)
the problem here is the term ln(1 + g). Is there any way to simplify this or if
not find some approximation to it? Fortunately there is. We have 3
ln(1 + g) ≈ g
(4)
so for a first approximation, we have
0.693 15 = gt
Where does 72 come from and why is this not simply called the rule of 69?
When the doubling proceeds quickly, instantaneously as in bacteria or nuclear
chemistry, then it is the rule of 69. (See The Story of e). In economics change
typically takes place more slowly, so we use the rule of 72 (sometimes 70) instead
of the instantaneous 69.
Example 8 The growth rate of the economy is 3.5%. How long will it take for
GDP to double? Answer: From the rule of 72, write
72/3.5 = t
or 20.571 years. As seen this is an approximation and will be a better approximation for smaller growth rates. 4
3 The
proof of why this works depends on a power series expansion
ln(1 + g) = g −
1 2
1
1
1
g + g 3 − g 4 + g 5 + ...
2
3
4
5
The approximation comes from taking the first term only!
4 To express the growth rate as a decimal, the rule becomes the rule of 0.72.
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Example 9 Why can we not take the yearly rates of growth of real GDP and
average them? Answer: Because of compounding. The growth in year t depends on the level in the previous year. But the level is the sum of the previous
level plus the growth in the previous period. So this year’s level is the sum of
the growth of the previous level plus the previous growth. This makes growth
geometric and not arithmetic. Here is what we mean. Let’s say that GDP is
100 in the base year. If it grows by 6 percent the first year and 12 percent the
second year, the average growth rate is 9 percent per year. This gives
[100 ∗ (1 + 0.09)](1 + 0.09) = 118.1
but in reality:
[100 ∗ (1 + 0.06)](1 + 0.12) = 118.72
a bit larger. So in general we must use equation 3 instead.
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